While testifying before the Senate Intelligence Committee on Thursday, Comey, in his opening statement, said the president had privately told him in the Oval Office, "I hope you can see your way clear to let this go, to letting Flynn go".
"I didn't say that", Trump said, which amounts to accusing Comey of perjury - or willfully lying after taking an oath to tell the truth.
Comey said he believed Trump fired him because of the Russian Federation investigation, and Trump himself cited "this Russian Federation thing" as a motivating factor for his decision to sack Comey during an interview with broadcaster NBC that aired last month.
The president tweeted a veiled threat at Comey days after he fired him, writing, "James Comey better hope that there are no "tapes" of our conversations before he starts leaking to the press!"
Gingrich said Comey represented the deep state "permanent opposition" to the president, who would stop at nothing to "keep their influence" and harm Trump's presidency.
When reporters kept shouting questions about the tapes, Trump shot back, "Oh you're going to be very disappointed when you hear the answer".
Apart from the questions about Comey, the two leaders pledged cooperation against terrorism amid a deepening diplomatic crisis in the Middle East.
The committee want the material by 23 June.
The leaders of the House Russia investigation, Reps. "Who would do that; it hardly makes sense", Trump said dismissively of Comey.
He also claimed that Mr Comey's comments to the Senate Intelligence Committee failed to prove any collusion between the Trump campaign and Russian Federation.
Records of Comey's private discussions with Trump took on renewed interest this week after Comey testified that Trump asked him to drop an investigation into former national security adviser Michael Flynn and implied Trump may have become the subject of investigation himself for interfering in the FBI's work.
Trump Jr. also told Pirro that Comey's testimony "vindicated" his father and proved that the former Federal Bureau of Investigation director is "a liar".
Read: Did James Comey Lie During Testimony?
Comey's claims in the memo have led to allegations from Democrats that Trump improperly interfered in the Flynn investigation and may have obstructed justice. "And some of the things he said just weren't true". He said he can also be interviewed by the special counsel who investigates Trump associates' ties to Russia, Robert Mueller.