This account is based on interviews with more than two dozen current and former Turkish and U.S. government officials, lobbyists and lawyers with direct knowledge of the interactions. Representatives for the Turkish government, Halkbank and the White House declined to comment.

Turkey had mounted an elaborate influence campaign in Washington to deal with Halkbank. It predated Mr. Trump’s election but came to encompass a broad cast of players, including Rudolph W. Giuliani, the former New York mayor; Michael T. Flynn, Mr. Trump’s first national security adviser; and Brian D. Ballard, a lobbyist and fund-raiser for the president.

After senior Turkish government officials lobbied Treasury Secretary Steven Mnuchin and Mr. Trump, Mr. Mnuchin pressed the Justice Department not to impose too large a fine on Halkbank because Turkey could not afford it, two federal officials said. Mr. Mnuchin’s office declined to comment on Halkbank but added that the Treasury and Justice Departments “routinely consult and coordinate” on sanctions cases and fines.

Mr. Bolton and others said they could not fully explain why Mr. Trump seemed so determined to please Mr. Erdogan.

“This was a relationship that was really important for the United States to handle,” said Fiona Hill, who oversaw policy on Turkey and Europe for the National Security Council under Mr. Trump. “And at every turn, the president kept leaping in, and he wasn’t following the strategic threads of the relationship.”

President Recep Tayyip Erdogan of Turkey, shown in a government handout photograph, addressing members of the Justice and Development Party this month. Mr. Erdogan pressed President Trump to quash a federal criminal investigation into Halkbank.

The Halkbank Campaign

Turkey’s lobbying campaign had started before Mr. Trump took office.

During a one-day visit to the country in August 2016 by Vice President Joseph R. Biden Jr., the Turkish president pulled Mr. Biden aside under a tree for a private conversation, according to an aide to the vice president on the trip.

The investigation of Halkbank, Mr. Erdogan claimed, was a “big conspiracy” instigated by his rival Fethullah Gulen, a charismatic Muslim cleric. Mr. Gulen left Turkey in the late 1990s and moved to Pennsylvania, where, in Mr. Erdogan’s telling, he plotted an unsuccessful coup attempt just a month earlier, according to a summary of the conversation provided to The Times by the Biden aide.

Mr. Erdogan asked Mr. Biden to remove Preet Bharara, then the U.S. attorney for the Southern District of New York. That office was in the early stages of an investigation into Halkbank and had already indicted a Turkish-Iranian gold trader, Reza Zarrab, for helping to orchestrate the sanctions-evasion scheme.

Mr. Erdogan also wanted the Obama administration to remove the judge overseeing Mr. Zarrab’s case in Manhattan, the Biden aide said. And he wanted Mr. Zarrab released and allowed to return to Turkey.

According to the Biden aide’s account, Mr. Erdogan said that if the United States really meant what it said about repairing relations, the case needed to go away.

Speaking to reporters before he left Turkey, Mr. Biden made clear that there were limits to what the United States could or should do to address Mr. Erdogan’s requests, including any effort to extradite Mr. Gulen.

“If the president were to take this into his own hands, what would happen would be he would be impeached for violating the separation of powers,” Mr. Biden said, with Mr. Erdogan at his side.

The Turkish president did not give up. He again raised Halkbank with Mr. Biden during a visit to New York for the United Nations General Assembly, and then twice in calls with President Barack Obama in the weeks before he left office in January 2017, aides to Mr. Obama and Mr. Biden said.

In an interview, Mr. Bharara said he never heard questions about the Halkbank investigation from anyone at the Obama White House or the attorney general’s office.

Mr. Trump’s election brought an immediate shift in Turkey’s outreach effort.

The Trump family had helped build Trump Towers Istanbul, a successful residential and retail complex that attracted Mr. Erdogan to its ribbon-cutting in 2012.

“I’ve gotten to know Turkey very well,” Mr. Trump said in 2015. “They’re amazing people, they’re incredible people. They have a strong leader.”

In Turkey, there was confidence that Mr. Erdogan’s agenda would now win attention and support at the highest levels of the U.S. administration.

“Top leadership in Turkey felt that Trump would be a tough-minded businessman, but a businessman they could work with,” Robert Amsterdam, a lobbyist for Turkey, recalled.

The investigation of Halkbank, Mr. Erdogan claimed, was a “big conspiracy” instigated by his rival Fethullah Gulen, who is living in the United States. The New York Times

Indictments and Testimony About Erdogan

Once Mr. Trump took office, there were early hints that Mr. Erdogan’s message was getting through to the White House.

The National Security Council asked the Education Department about a network of charter schools, partly funded with federal money, that were said to be linked to Mr. Gulen, the Erdogan rival who was living in Pennsylvania. The agency was then asked if the money could be blocked, one official involved in the conversations said. But Education Department officials resisted, saying they did not have the legal authority to stop the funding.

The White House encountered a similar lack of enthusiasm at the Department of Homeland Security and the F.B.I., which received requests to investigate Mr. Gulen and look for ways to perhaps force him out of the United States, officials involved in the efforts said. The F.B.I. declined to comment.

In the meantime, the effort by Turkey to resolve the Halkbank case intensified.

Records show that the bank and the Turkish government paid Mr. Ballard’s lobbying firm $4.6 million over two years for work on Halkbank and other matters, including meetings and phone calls with the vice president’s office, the State Department, members of Congress and Jay Sekulow, one of Mr. Trump’s personal lawyers.

Mr. Ballard’s team argued that the Halkbank case was a foreign policy matter, and that the need to maintain close relations with Turkey, a NATO member, had to be taken into account.

But the investigation by the federal prosecutors in Manhattan ground ahead. By early 2018, it had led to the indictments of nine defendants, including Turkey’s former economy minister and three Halkbank officials, on charges such as bank fraud and money laundering related to the sanctions-evasion scheme.

One defendant, Mehmet Hakan Atilla, the bank’s deputy general manager for international banking, was tried and, in January 2018, convicted.

Mr. Zarrab, the gold trader, had pleaded guilty and testified about how the scheme had relied on false documents and front companies, and how he had paid millions of dollars in bribes to the economy minister and Halkbank’s general manager.

He also testified that the operation had Mr. Erdogan’s knowledge and approval, as well as that of Mr. Erdogan’s son-in-law, Berat Albayrak, who now serves as Turkey’s finance minister.

When Mr. Berman was appointed U.S. attorney in January 2018, prosecutors were turning their focus to the bank itself and the possibility of charging others involved in the scheme.

Halkbank’s lawyers held repeated talks in 2018 with Mr. Berman’s office over whether a “global settlement” could be reached. But the two sides were far apart, according to people briefed on the meetings.

The prosecutors said they were prepared to allow the bank to avoid indictment if it agreed to pay a heavy fine, reform its operations and make a series of admissions about its conduct, the people who were briefed said.

But the bank’s lawyers argued that Halkbank and its executives had done nothing wrong, that they had been deceived by Mr. Zarrab, and that his testimony was untrue. They said the bank would not make the required admissions.

There were indications by then that Turkey’s arguments were being heard in Washington.

At the Treasury Department, Mr. Mnuchin considered the violations Halkbank had been accused of to be serious — and he believed the U.S. government was right to demand that the bank admit wrongdoing, according to one White House official involved in the negotiations.

But Mr. Mnuchin raised concerns about how large a fine might be imposed on Halkbank. The French banking giant Société Générale agreed that same year to pay U.S. authorities more than $2 billion to resolve charges that it had violated U.S. sanctions against Cuba and bribed officials in Libya, among other accusations.

A fine on that scale would threaten the future of Halkbank, lobbyists and lawyers for the bank argued, as did top Turkish officials in conversations with members of the Trump administration. One direct appeal to Mr. Mnuchin came from Mr. Albayrak, Mr. Erdogan’s son-in-law.

In 2018, Mr. Mnuchin reached out about the scale of a potential fine to Jeff Sessions, the attorney general at the time. Justice Department officials then asked Southern District prosecutors whether the size of the fine they were demanding was negotiable, one lawyer involved in the effort said. The response was affirmative: The amount was less important than securing an admission of wrongdoing.

“We need an admission of liability” was the message the Manhattan prosecutors sent back to Washington, according to two lawyers involved in the matter.

Mr. Trump and Mr. Erdogan at the NATO summit meeting in Brussels in 2018. Doug Mills/The New York Times

‘Well, It Looks Convincing to Me’

On the second day of a trip to Buenos Aires in late 2018 for the annual Group of 20 gathering of world leaders, Mr. Trump met with Mr. Erdogan for talks intended to be focused on issues like continued tensions over Islamic State operations in Syria.

But the conversation quickly went off course.

Mr. Erdogan made clear that he was frustrated with the continued pestering by Southern District prosecutors concerning Halkbank, and he wanted Mr. Trump to intervene to help wrap up the investigation, Mr. Bolton said in the interview.

Mr. Erdogan handed Mr. Trump a copy of a memo written by Halkbank’s lawyers explaining why Turkey believed the Justice Department had misconstrued U.S. sanctions law. It argued that Halkbank’s trading with Iran was not illegal, because it was largely based on trades of gold and food that were not in dollars and did not involve U.S. banks.

Mr. Trump flipped through the memo quickly, Mr. Bolton said.

“Well, it looks convincing to me,” Mr. Trump said, according to Mr. Bolton, who also recounted the meeting in his recent book.

By Mr. Bolton’s account, Mr. Trump also told Mr. Erdogan that he wanted to replace the prosecutors in Mr. Berman’s office in Manhattan, whom Mr. Trump considered to be holdovers from the Obama era.

Two weeks later, in mid-December 2018, Mr. Trump and Mr. Erdogan spoke by phone. The president began by assuring Mr. Erdogan that the government and Halkbank were close to a resolution, and Mr. Erdogan expressed his appreciation, according to Mr. Bolton.

In Turkey around this time, Mr. Erdogan told reporters that Mr. Trump, in an earlier conversation about Halkbank, had assured him that Mr. Trump “would instruct the relevant ministers immediately” to take care of the matter.

Mr. Bolton said in the interview that his concern, as he listened to these conversations, was that Turkey and Halkbank now “had a direct channel in the Oval Office — they weren’t going to negotiate in good faith” with the prosecutors. “Why should they?”

Mr. Trump asked Mr. Bolton to speak with Mr. Whitaker, the acting attorney general at the time, about the case — a move Mr. Bolton said he did not make, although he added that he did not know if someone else from the White House did.

On Dec. 14, the day of the telephone call between Mr. Erdogan and Mr. Trump, the Justice Department notified the Southern District that Mr. Mnuchin, Secretary of State Mike Pompeo and the attorney general’s office would become more involved in the Halkbank case, one Justice Department official said.

The prosecutors in Manhattan had just drafted a memo for Mr. Whitaker and Rod J. Rosenstein, the deputy attorney general, detailing why the Justice Department should give them the authority to file criminal charges against the bank, two lawyers said.

Mr. Rosenstein was convinced that the evidence was compelling, perhaps even more so than in other sanctions-evasion cases in which the United States had charged banks, lawyers familiar with the investigation said. The memo from the prosecutors also noted that the actions Halkbank was accused of taking were helping to support Iran’s economy, which was antithetical to Mr. Trump’s foreign policy goal of tightening economic pressure on the country.

Mr. Rosenstein urged Mr. Berman to come to Washington to present the Southern District’s argument to Mr. Whitaker. The goal was not to file charges immediately against the bank. Instead, the plan was to give the Southern District more leverage to squeeze Halkbank to accept a deferred prosecution agreement that included an admission of wrongdoing.

But Mr. Whitaker, who declined requests for comment, had a longstanding disdain for the Southern District, which has been called the Sovereign District for the way it guards its independence from Washington. In a book published this year, Mr. Whitaker wrote that the Southern District always “dreamed up new ways to torment President Trump throughout my tenure at the Department of Justice.”

Mr. Berman arrived at the Justice Department headquarters and reported to Mr. Rosenstein’s office. But shortly before the meeting was to begin, Mr. Rosenstein was summoned to Mr. Whitaker’s office without Mr. Berman.

Mr. Whitaker told Mr. Rosenstein that he did not want the case to move forward, and that he wanted the matter shut down, according to lawyers involved in the investigation. Mr. Whitaker cited concern that charges against the bank might result in a threat to U.S. forces in Syria, a suggestion that others in the department said they found hard to understand.

Justice Department officials decided to ignore Mr. Whitaker’s edict, concluding that they most likely would outlast Mr. Whitaker in the department, since he was serving on an acting basis. They did not see appeasing Mr. Erdogan as sufficient justification for closing the investigation.

Attorney General William P. Barr proposed a “global settlement” that would end the inquiry into Halkbank. Federal prosecutors in Manhattan objected. Doug Mills/The New York Times

‘This Is Not How We Do Things at the Southern District’

Mr. Barr was confirmed as the new attorney general in mid-February 2019, a few months after Mr. Whitaker had pushed to end the case. The prosecutors in Manhattan were encouraged that they might now get the charging authority they wanted.

But Mr. Erdogan and his top advisers continued to lobby Mr. Trump and members of his cabinet, including Mr. Mnuchin and now Mr. Barr.

One of the appeals came from Mehmet Ali Yalcindag, a Trump family friend who had been closely involved in developing the Trump towers in Turkey and who now leads a Turkey-U.S. business trade group. On a trip to Washington that April, he pressed administration officials about the bank.

Discussions between Halkbank and the Southern District continued, according to lawyers involved in the case. But the bank maintained its refusal to admit to wrongdoing and insisted on a deal that would end investigations and drop existing charges.

At times, the prosecutors were left with the impression that bank officials felt they had all the leverage because of the relationship between Mr. Trump and Mr. Erdogan.

In mid-June 2019, when Mr. Berman met with Mr. Barr in Washington, the attorney general pushed Mr. Berman to agree to allow the Justice Department to drop charges against the defendants and terminate investigations of other suspected conspirators, according to a former department lawyer familiar with the session.

Among the defendants with charges pending were Halkbank’s former general manager, Suleyman Aslan, and Turkey’s former economy minister, Mehmet Zafer Caglayan.

The suggestion that the Justice Department would offer Turkish officials protection from criminal charges, even without their agreement to assist in the investigation, was unacceptable and unethical, Mr. Berman argued, according to lawyers close to the investigation. Justice Department policy specifically says that criminal conduct by individuals is not resolved when a company admits wrongdoing.

“This is not how we do things at the Southern District,” Mr. Berman told Mr. Barr, adding that he would not agree to such a move and that his office would not be part of it.

Mr. Barr sought to persuade Mr. Berman that the so-called global settlement would enforce U.S. sanctions law and avert a rift with an ally in a volatile part of the world.

Aykan Erdemir, a former member of Turkey’s Parliament and a critic of Mr. Erdogan, who was not part of the negotiations, said such a proposal by Mr. Barr would be a gift to Mr. Erdogan and critical to his political standing in Turkey by eliminating potential criminal charges against members of his inner circle.

“That is the biggest prize that Erdogan could ever receive,” Mr. Erdemir said. “Erdogan was not trying to save the bank. He was trying to save his ministers and save himself.”

The negotiations had reached an impasse. Mr. Barr had the power to stop any new criminal charges. But to dismiss any existing cases, the federal prosecutors in Manhattan would need to seek judicial approval.

Lawyers in the Justice Department’s national security division took over the negotiations, but they also ended up frustrated, people briefed on the matter said.

In his recent book, Mr. Bolton said he had warned Mr. Barr in April 2019 about Mr. Trump’s penchant to “give personal favors to dictators.”

In the interview with The Times, Mr. Bolton said he did not know the details of Mr. Barr’s intervention in the Halkbank negotiations. But he said he was disturbed by the tenor of the interaction between Mr. Trump and Mr. Erdogan related to Halkbank.

“It was so idiosyncratic, so personal to Trump in the pursuit of personal relationships, that it was very dangerous,” Mr. Bolton said. “And it does look like obstruction of justice.”

Just how idiosyncratic became more apparent last October, when Mr. Erdogan sent troops into Syria. Mr. Trump, who had initially given Mr. Erdogan the green light to do so, then faced an intense bipartisan backlash, leading him within days to take a tougher line with Turkey, threatening economic reprisals.

“You don’t want to be responsible for slaughtering thousands of people, and I don’t want to be responsible for destroying the Turkish economy — and I will,” Mr. Trump wrote to the Turkish leader on Oct. 9, 2019, without elaborating.

On Oct. 15, the Justice Department gave the prosecutors in Manhattan approval to file charges against Halkbank, a direct slap at Mr. Erdogan.

The prosecutors rushed to present evidence before a grand jury and secured a six-count indictment that same day charging Halkbank with money laundering, bank fraud and conspiracy to violate the Iran sanctions. So far, no additional individuals have been charged.

When the charges against the bank were announced, Mr. Berman said in a statement that the “bank’s audacious conduct was supported and protected by high-ranking Turkish government officials, some of whom received millions of dollars in bribes to promote and protect the scheme.”

In June, eight months after the indictment was returned, Mr. Trump fired Mr. Berman. Justice Department officials cited his handling of the Halkbank matter, including his blocking of the proposed global settlement, as a key reason for his removal.

Katie Benner and Adam Goldman contributed reporting.

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Eric Lipton is a Washington-based investigative reporter. A three-time winner of the Pulitzer Prize, he previously worked at The Washington Post and The Hartford Courant.

Benjamin Weiser is a reporter covering the Manhattan federal courts. He has long covered criminal justice, both as a beat and investigative reporter. Before joining The Times in 1997, he worked at The Washington Post.

A version of this article appears in print on Oct. 30, 2020, Section A, Page 1 of the New York edition with the headline: Trump Ties to Erdogan Snarled U.S. Inquiry Into Turkish Bank.