Trump abruptly fired Comey last week, saying he did so based on his very public handling of the Hillary Clinton email probe.
But the White House has provided differing accounts of the firing. And lawmakers have alleged that the sudden ouster was an attempt to stifle the bureau's investigation into Trump associates' ties to Russia's alleged meddling in the 2016 presidential election.
Comey's memo detailing his conversation with Trump would be the clearest proof to date that the president has tried to influence that investigation. The Times said it was part of a paper trail Comey created documenting what he saw as Trump's efforts to improperly interfere in the ongoing probe.
The Justice Department declined to comment.
According to the Times, Comey wrote in the February memo that Trump told him Flynn had done nothing wrong. But Comey did not say anything to Trump about limiting the investigation, replying, "I agree he is a good guy."
The newspaper said Comey was in the Oval Office that day with other national security officials for a terrorism threat briefing. When that ended, Trump asked everyone to leave except Comey, and he eventually turned the conversation to Flynn.
On Tuesday, for the second night in a row, Senate Republicans and Democrats were caught off-guard as they entered the chamber for a scheduled vote.
"I don't know the facts, so I really want to wait until I find out what the facts are before commenting," Sen. John Cornyn, R-Texas, told reporters.
Asked if it would be obstructing justice for Trump to have asked Comey to drop the Flynn investigation, Cornyn said: "You know, that's a very serious charge. I wouldn't want to answer a hypothetical question."
Rep. Jason Chaffetz, Republican chairman of the House intelligence committee, said he was ready to subpoena the memo, which was an apparent effort to create a paper trail of Comey's contacts with the White House.