He is Senate majority whip, making him the second-ranking Republican in the chamber.
A source close to Comey tells @NBCNews: "He hopes there are tapes".
Sources told ABC News that Trump posed the question to Comey a few times during a dinner in January and Comey would not promise such a thing to the new president, only that he would be honest.
The U.S. intelligence community previously concluded that the Russian government interfered in last year's White House race by using state-sponsored hackers to infiltrate Democratic targets and steal sensitive correspondence later published online by WikiLeaks and others prior to Mr. Trump's election.
"Even that is possible", Mr Trump said.
On Tuesday, when Comey's dismissal was announced, the administration suggested that Trump was acting on the recommendation of Atty. Gen.
"President Trump and the White House have presented an ever-changing narrative on the rationale for the firing of FBI Director Comey", Feinstein said in a statement.
At the beginning, Trump's aides said Comey's firing came at the suggestion of Attorney General Jeff Sessions and Deputy Attorney General Rod J. Rosenstein, but by Wednesday the word was that Trump had been considering firing Comey for a long time, and made the final determination after hearing from Sessions and Rosenstein.
The BBC's Laura Bicker in Washington says Mr Trump was referring to a private White House dinner in January during which the president claims Mr Comey assured him he was not being investigated over alleged links to Russian Federation. "I'm not aware of that", Spicer said. "I think it's a good idea", Trump said.
The president said he never pressured Comey into dropping the Federal Bureau of Investigation probe, and added that there was no "collusion between me and my campaign and the Russians". Trump said in excerpts of an NBC News interview released Friday that Comey "should have never exonerated" Hillary Clinton.
The 49-year-old McCabe, a career FBI agent, has taken part in a number of high-profile investigations, including probes into the Boston Marathon bombing in 2013 and the attacks against U.S. installations in Benghazi, Libya in 2012. Comey instead told Trump he could count on his honesty, the Times said.
Senior Democrats on the House judiciary and oversight committees wrote to the White House on Friday demanding copies of any recordings.