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.