Speaking to Lester Holt on NBC News on Thursday Trump gave a different account to that of his team.
Trump sparked a political firestorm when he abruptly fired Comey last week.
He also threatened to cancel press briefings, suggesting via Twitter that they might instead be replaced with written statements "for the sake of accuracy".
Senate Minority Leader Chuck Schumer of NY said the new Federal Bureau of Investigation director should be someone "not of partisan background" with "great experience" and "courage". He dismissed as less desirable at least two of the 14 candidates under consideration, ex-FBI agent and former Rep. Mike Rogers of MI and Sen.
Mr Rosenstein's letter says "the FBI's reputation and credibility have suffered substantial damage, and it has affected the entire Department of Justice". Of those interviewed, 40 percent identified themselves as voters who cast ballots for Democratic candidate Hillary Clinton previous year while 37 percent said they voted for Trump.
Rogers was endorsed over the weekend by the FBI Agents Association, which represents rank and file agents.
"I can not defend the Director's handling of the conclusion of the investigation of Secretary Clinton's emails, and I do not understand his refusal to accept the almost universal judgment that he was mistaken", Mr Rosenstein goes on to say.
The president insinuated in a tweet last week that he has tapes of conversations he held with Comey.
Plenty of people were all abuzz over Donald Trump's suggestion that he might be taping some White House conversations. But Trump then said Thursday he had planned to fire Comey regardless. When asked whether the Russian probe played a part in Comey's firing, Pence said on Wednesday "that's not what this is about".
Pence had the poor judgment to argue that Trump had chose to fire FBI Director Jim Comey only on the recommendation of Deputy Attorney General Rod Rosenstein and Attorney General Jeff Sessions. "The public has a right to know why Comey was sacked so the president can be held accountable for any abuse of his position".
"The key here, of course, is getting some of our Republican colleagues to join us", Schumer said. He said Comey asked Trump to retain him as FBI director and assured the president he was not under investigation.
Administration sources told The Daily Beast that the FBI's associate deputy director, David Bowdich, could be pushed out after playing an important role in the ongoing investigation that forced Flynn's resignation. Ben Sasse, a Nebraska Republican, told CBS News on Monday. Trump doesn't seem to care very much about subcabinet nominations going through rapidly, and the legislative agenda just isn't very heavy, meaning that demands on Senate floor time are less than usual.
The only in-person instance Trump described was a conversation at a "nice dinner" in which the president's status was discussed. "You can't be cute about tapes".
And the American people would be better off if reporters could spend less of their time distracted by the White House press aides and more of their time digging into what's actually going on in the administration.
On Capitol Hill, where the House will return Monday after a recess, members continued to disagree about the need for a special prosecutor to take over the Russian Federation investigation.
Republican Sen. Mike Lee of Utah said on "Fox News Sunday" that if there are recordings, it is "inevitable" that they will be subpoenaed and have to be turned over.
"He viewed Mr Comey as eager to step in front of TV cameras and questioned whether his expanding media profile was warping his view of the Russian Federation investigation, the officials said". I won't talk about that. "All I want is for Comey to be honest, and I hope he will be, and I'm sure he will be, I hope".
The Washington Post reported on Saturday that Trump regularly recorded his conversations as a businessman.
Any nominee must be confirmed by the Senate.style="text-align: center;"