The source close to Comey told NBC News that Trump told the then-FBI chief in a White House meeting: "I hope you can see your way clear to letting this go, to letting Flynn go".
The White House issued an immediate denial, saying the report is "not a truthful or accurate portrayal of the conversation between the president and Mr Comey".
Comey didn't directly respond to Trump's request, which was made in February a day after Flynn was sacked for deceiving the vice president about his contacts with Russia's ambassador to the USA, said the person, who spoke on condition of anonymity.
Those same people told The Washington Post that the details of those notes were shared with a small group of people at the Federal Bureau of Investigation and Justice Department. But the White House has provided differing accounts of the firing. Obviously, Trump had knowledge that Flynn was the target of an FBI investigation. The memo was part of a paper trail Comey created documenting what he perceived as the president's improper efforts to influence an ongoing investigation.
The associate who described the memo said Comey is willing to testify but wants to do it in public to ensure a full airing of events.
Several lawmakers and former Obama aides have raised concerns of obstruction of justice.More news: Manhunt for hackers behind global cyberattack
Sen. Mark Warner of Virginia, the top Democrat on the Senate Intelligence Committee, said he will ask Comey for additional material as part of the panel's investigation.
"If these reports are true, the President's brazen attempt to shut down the FBI's investigation of Michael Flynn is an assault on the rule of law that is fundamental to our democracy", House Minority Leader Nancy Pelosi said in a statement.
The Times said two unnamed people read the memo written by Comey after he had a conservation with Trump after an Oval Office meeting in February.
Comey did not say anything to Trump about curtailing the investigation, only replying: "I agree he is a good guy".
The White House vigorously denied it all. "Memos, transcripts, tapes - the list keeps getting longer", he said. He called for the appointment of a special prosecutor to investigate. He said Comey will likely come back to testify in the investigation. When the meeting ended, Trump told those present - including Pence and Attorney General Jeff Sessions - to leave the room except for Comey.
The administration spent the first half of Tuesday defending Trump's disclosure of classified information to senior Russian officials. Chaffetz's letter to acting FBI Director Andrew McCabe requests all memoranda, notes, summaries and recordings that relate to any communications between Comey and the president.