WASHINGTON — It was late 2018, and President Recep Tayyip Erdogan of Turkey was on the phone with an unusual request for President Trump: Could he intervene with top members of his cabinet to curb or even shut down a criminal investigation into Halkbank, one of Turkey’s largest state-owned banks?

It was not Mr. Erdogan’s only effort to persuade the Trump administration to back off the investigation into the bank, which had been accused of violating United States sanctions against Iran.

His government had hired a lobbying firm run by a friend of and fund-raiser for Mr. Trump to press his case with the White House and State Department. And there would be more phone calls between the two leaders in which the topic came up, according to participants in the lobbying.

Mr. Erdogan’s influence campaign is now under scrutiny again in Washington, following the disclosure that Mr. Trump’s former national security adviser, John R. Bolton, reported in his forthcoming book his concern that the president was effectively granting personal favors to Mr. Erdogan and President Xi Jinping of China.

People familiar with the unpublished manuscript said Mr. Bolton wrote that he had shared his concern with Attorney General William P. Barr and that Mr. Barr responded by pointing to Mr. Trump’s intervention in two cases linked to Turkey and China: the investigation of Halkbank and Mr. Trump’s decision in 2018 to lift sanctions on ZTE, a major Chinese telecommunications company.

The Justice Department has disputed Mr. Bolton’s account. But on Tuesday, top Democrats seized on the suggestions of meddling in the Halkbank and ZTE cases as fresh evidence that Mr. Trump, whose family enterprise has extensive business ties to Turkey and also has considered building new towers in China and expanding in other areas, was using the presidency to enrich himself and his family.

“Several members of the administration had concerns about the president’s dealings with autocrats,” Senator Chuck Schumer of New York, the Democratic minority leader, said at a news conference. “Did the president have financial interests at stake when he was talking to Erdogan or Xi and others?”

He added: “Maybe his kids had some economic interest at stake. And did it impact our nation’s foreign policy with those countries?”

Former foreign policy officials — including some who served in Republican administrations — said in interviews that Mr. Trump plays an unusual and at times disturbing role in high-profile criminal and sanction cases involving foreign governments.

“What I know about his intervention in the Halkbank case is highly abnormal and quite worrying, actually,” said Philip Zelikow, a history professor at the University of Virginia who served on the National Security Council staff for President George Bush.

Suggesting that Mr. Trump was putting private, commercial interests above those of the United States, Mr. Zelikow added: “There have been interventions on behalf of a foreign government that are hard to explain in traditional public interest terms.”

Mr. Trump’s involvement in the Halkbank investigation started early in his administration. In 2017, he was asked by Rudolph W. Giuliani during an Oval Office meeting with Rex W. Tillerson, then the secretary of state, to help secure the release of a Turkish gold trader at the center of Halkbank’s sanctions-evasion efforts.

The gold trader, Reza Zarrab, who had hired Mr. Giuliani to help secure his release, had been accused by federal prosecutors of playing a central role in an effort by Halkbank to funnel more than $10 billion in gold and cash to Iran, in defiance of United States sanctions designed to curb Iran’s nuclear program.

Turkey also wanted the trader released, former Turkish government officials said, so that he would not testify against top bank officials or implicate members of Mr. Erdogan’s family or Mr. Erdogan himself.

But that was just the start of the lobbying.

Mr. Erdogan, in a series of phone calls and in-person conversations in 2018 and 2019, repeatedly tried to persuade Mr. Trump to use his power to limit additional enforcement action against Halkbank itself, something the Justice Department had made clear it was considering.

After one phone conversation in late 2018, Mr. Erdogan told reporters in Turkey that Mr. Trump had told him that “he would instruct the relevant ministers immediately” to follow through on the matter.

“Talks are underway about this issue,” Mr. Erdogan said at the time. “It is very important that this process has begun.”

Mr. Erdogan’s son-in-law, who serves as Turkey’s finance minister, also took up the case, pressing Treasury Secretary Steven Mnuchin on the matter. Other appeals were made by the former project manager of Trump Towers Istanbul, a twin-tower complex and mall that was the Trump family’s first high-rise project in Europe.

Asked about Turkey’s lobbying efforts in an interview in October, Mr. Mnuchin cited the ongoing legal process and would not comment.

The bank had separately hired a lobbying firm run by Brian D. Ballard, a top fund-raiser for Mr. Trump’s campaign and the Republican National Committee. The lobbyists from Mr. Ballard’s team argued to the State Department and White House that any criminal charges against the state-owed bank could destabilize the Turkish economy.

For months, it looked like Turkey was going to succeed in this unusual lobbying campaign — asking a United States president to put pressure on his own Justice Department to protect a state-owned bank. Mr. Barr, who was confirmed in February 2019, played a key role in overseeing the negotiations over a possible settlement with the bank that would have seen it avoid criminal charges, representatives for Halkbank said in interviews last year.

Only after Turkey invaded Syria in early October did the Justice Department move to indict the bank.

“President Trump has been carrying water for President Erdogan and Turkey’s state-owned Halkbank,” said Senator Ron Wyden of Oregon, the top Democrat on the finance committee. “Every member of Congress should be profoundly alarmed that Donald Trump is trying to get the bank accused of the largest Iranian sanctions violation scheme in U.S. history off the hook because his authoritarian pal asked for a favor.”

Mr. Trump’s 2018 intervention in the case of ZTE was equally perplexing to some observers. Two years before, the United States found the Chinese company guilty of violating American sanctions on Iran and North Korea. In April 2018, the Trump administration moved to punish ZTE by banning it from buying American technology.

But Mr. Trump suddenly had a change of heart, essentially pardoning the company in exchange for a $1 billion fine and promises to replace its senior leadership and allow American compliance monitors.

The decision came after a direct plea to Mr. Trump from Mr. Xi in the midst of intense maneuvering over trade talks between the two countries and as the United States was preparing for a summit with North Korea.

It drew bipartisan backlash on Capitol Hill. Top lawmakers, including Senators Marco Rubio of Florida, Tom Cotton of Arkansas and Chris Van Hollen of Maryland, had urged the administration not to bend on ZTE, which they considered a law enforcement and national security issue.

Chinese officials had made it clear that they considered lifting ZTE’s penalty a condition for reaching a trade deal. There was also the implicit threat that, if the penalty was not lifted, American companies operating in China would face further retaliation. The United States has also relied on China to exert pressure on North Korea to dismantle its nuclear program.

The Trump family had for years worked on plans to build a series of new hotel or apartment building projects in China, goals put on hold after Mr. Trump was elected president.

His administration scrambled to quiet the growing dissent, and Mr. Trump lashed out at Democrats for having allowed ZTE to flourish under President Barack Obama’s watch.

In May 2018, Wilbur L. Ross Jr., the commerce secretary, and Mr. Mnuchin traveled to Capitol Hill to brief a group of Senate Republicans, including Mr. Rubio, John Cornyn of Texas and Bob Corker of Tennessee, on their plans for ZTE. Mr. Ross and Mr. Mnuchin sought to assure the lawmakers that they were planning on harsh penalties for ZTE and appealed to Republicans to dampen their public criticism so a deal could be reached, a person briefed on the discussions said.

Chinese officials widely speculated that the penalties on ZTE were an effort by the Trump administration to gain the upper hand in the trade talks. But people briefed on the discussions say Trump administration officials had not fully realized what a complication the measure would become in the trade talks.

Since then, ZTE has made a gradual recovery, and its profits have rebounded. And although its run-in with the Trump administration tarnished its smartphone brand with consumers, cellphone carriers around the world have still been willing to work with the company to build 5G mobile networks.

The handling of ZTE has raised questions about whether Mr. Trump will follow through with imposing restrictions on Huawei, another Chinese telecommunications company that the White House views as a national security threat.

Elizabeth Rosenberg, who served as a senior Treasury adviser working on sanctions issues during the Obama administration and who now studies sanctions policy at the Center for a New American Security, said Mr. Trump’s interventions were unusual and disruptive.

“This is not the norm in Washington,” she said. “He is making up sanctions policy on his own, and influencing the course of policy in a way that undermines United States priorities and has shocked United States allies.”

Ana Swanson contributed reporting from Washington, Raymond Zhong from Shanghai and Michael Forsythe from New York.