She also defended the use of word "showboat" for Comey by Trump in an interview earlier in the day.
It was unclear whether Mr Trump's reference to "tapes" suggested there might be secret recordings of conversations that could be used to challenge any statements by Mr Comey, or whether it was simply a way of warding him off from commenting.
Also read: Here are the unanswered questions about Trump's "tapes" warning to Comey. Officials did not immediately respond to questions about whether Trump recorded his discussions with the Federal Bureau of Investigation director.
White House Deputy Press Secretary Sarah Huckabee Sanders said Wednesday that Trump fired Comey only after getting a recommendation from the Department of Justice.
The FBI probe and parallel congressional investigations have hung over Trump's presidency since he took office in January and threaten to overwhelm his policy priorities.
Trump's advisers have repeatedly tried to downplay the Russia-election matter, with Sanders saying Wednesday the FBI was "doing a whole lot more than the Russian Federation investigation". Luttig clashed with the Bush White House on a prominent terror case, rebuking the administration for its actions in the case involving "enemy combatant" Jose Padilla. "It is also key to allowing there to be transparency at the highest levels of government, whether that's the president of the United States or the people who work for him". Comey found the allegations confounding, according to his associate, and wondered what to make of what he described as odd thoughts coming from his new boss. "Jim Comey deserves his chance to lay out to the American public his side of the facts".
The No. 2 Democrat in the Senate, Richard Durbin, slammed the President after Friday's tweets, telling MSNBC that the President is "dangerous because he may be obstructing justice in terms of the investigation. and secondly his credibility has been destroyed".
The leading Democrats on the House judiciary and oversight committees asked the White House Counsel on Friday afternoon for copies of any recordings he might have. Today, the president tweeted out that he felt, from behind that podium, it's not always possible to present the information with ideal accuracy.
"Maybe the best thing to do would be to cancel all future "press briefings" and hand out written responses for the sake of accuracy???" added Trump on Twitter.
Mr Trump has said that Mr Comey has told him three times that he was not a target of the FBI inquiry, comments that have raised accusations among opponents that the president was interfering in the investigation. It said that Trump knew Comey was not up to the task so he chose to replace him with someone who could bring credibility back to the investigative agency.
"The president obviously wants to make sure that we've got the right person", Spicer said.
"I think, in many ways, our institutions are under assault, both externally - and that's the big news here, is the Russian interference in our election system", said James Clapper, the former director of national intelligence. The White House initially cited a Justice Department memo criticizing Comey's handling of last year's investigation into Hillary Clinton's emails as the impetus.