In testimony on Capitol Hill last week, Mr. Comey said Mr. Mueller had a copy of that memo and several others Mr. Comey had written about his interactions with Mr. Trump.

A spokeswoman for the White House referred all questions on the matter to Mr. Trump’s outside lawyer, Marc E. Kasowitz. A spokesman for Mr. Kasowitz said, “The F.B.I. leak of information regarding the president is outrageous, inexcusable and illegal.”

The scrutiny of Mr. Trump’s actions is part of a ripple of unintended consequences that began when the president, frustrated by the cloud of investigations into Russian collusion, fired Mr. Comey last month. “When I decided to just do it, I said to myself — I said, ‘You know, this Russia thing with Trump and Russia is a made-up story,’” Mr. Trump told NBC. He then said: “I might even lengthen out the investigation, but I have to do the right thing for the American people. He’s the wrong man for that position.”

The White House could try to assert executive privilege to keep the intelligence officials from discussing conversations between them and the president with Mr. Mueller. But that could set up a fight in court, where judges have generally held that criminal investigators can demand information that would normally be privileged.

In his memos, Mr. Comey said Mr. Trump had encouraged him to end an F.B.I. investigation into Mr. Flynn, an effort that Mr. Comey called “very disturbing.” There is a broad federal inquiry underway into Mr. Flynn’s actions. Among the issues being examined are whether he misled investigators about his ties to Russia, and his failure to disclose that he was working as a foreign agent of Turkey from August to November 2016: the same time he was advising the Trump campaign.

The Justice Department appointed Mr. Mueller last month to investigate whether members of the Trump campaign colluded with Russian operatives to influence the outcome of last year’s presidential election. Mr. Mueller inherited the criminal investigations into Mr. Flynn and Mr. Trump’s former campaign chairman, Paul Manafort. He was also given the authority to investigate obstruction.

While Rod J. Rosenstein, the deputy attorney general, has not said what exactly prompted him to appoint Mr. Mueller, his decision came after The New York Times published details about an Oval Office meeting Mr. Comey had with the president at the White House in February. During the meeting, the president brought up Mr. Flynn and told Mr. Comey, “I hope you can let this go,” according to the memo. Mr. Comey told the Senate that he viewed that as a clear directive from the president to drop the investigation.

A former senior official said Mr. Mueller’s investigation was looking at money laundering by Trump associates. The suspicion is that any cooperation with Russian officials would most likely have been in exchange for some kind of financial payoff, and that there would have been an effort to hide the payments, probably by routing them through offshore banking centers.