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.
"The challenge they have is that the president sometimes moves so rapidly that they don't get a team around that gets it organised", said Newt Gingrich, the former US House speaker and Trump ally.
Other interviews were expected with Adam Lee, the special agent in charge of the FBI's Richmond, Virginia, office; acting FBI Director Andrew McCabe; New York Court of Appeals Associate Judge Michael Garcia; and Senator John Cornyn, a Texas Republican, according to sources familiar with the search.
WikiLeaks made its offer on its Twitter site Friday after Trump warned in a tweet that Comey had "better hope" that the president didn't have tapes of their dinner meeting. He said most are "very well known" and of the "highest level".
The first interviews to find a replacement for fired FBI Director James Comey take place on Saturday. "You have these tremendous - the level of hatred toward him, especially during that period of time with Hillary Clinton where he gave her a free ride", Trump said. Trump told NBC on Thursday that he was looking for an excuse to fire Comey, who was leading an investigation into the Trump campaign's ties to Russian Federation.
Fisher, who would be the first woman to lead the Federal Bureau of Investigation if selected, was the first of the candidates to be interviewed, pulling into the Justice Department in a BMW just before 8 a.m. and leaving a little more than an hour later.
Triple Crown in sight as Always Dreaming tackles Preakness Stakes
The No. 6 post is the most successful starting gate spot, with 16 all-time Preakness winners (15.5 percent of all 106 entries). He's a really good 3-year-old and he's had a little bit more time to recover from the Arkansas Derby. "We got clobbered.
The White House had no immediate comment on Clapper's remarks on a morning in which no White House aide appeared on the Sunday news shows to discuss Trump's firing.
Fisher was assistant attorney general for the department's criminal division under President George W. Bush.
On Wednesday, Trump said Comey was let go because he "wasn't doing a good job".
He deliberately kept them in the dark because Trump was afraid the news would leak, which - to be perfectly fair - it absolutely would have.
In an interview with United States media, President Trump also accused him being a "showboat" and a "grandstander". In announcing the firing, the White House circulates a scathing memo, written by Deputy Attorney General Rod Rosenstein, criticizing Comey's handling of the Clinton probe.