The potential successors are Andrew McCabe, the acting FBI Director; Alice Fisher, a defence lawyer who used to lead the department's criminal division; Michael Garcia, a NY state appeals court judge; and John Cornyn, a Republican senator from Texas.
In the aftermath of Comey's dismissal, Cornyn said Trump was "within his authority" to fire him and said it would not affect the investigation of possible Russian ties to Trump's presidential campaign.
Trump says Comey had told him three times he was not under investigation in the Russian Federation probe.
Graham said that if Trump has any tapes of conversations with Comey, "they have to be turned over".
Mr Spicer returned at the end of the week for a briefing to a barrage of questions about Mr Trump's erratic tweets and one reporter to asked if the president was "out of control".
He said in an interview with NBC News he had already made a decision to sack Mr Comey by the time Deputy Attorney General Rod Rosenstein's memo came to him. "That would be flawless".
In two subsequent phone conversations, Comey also assured him he was not under investigation.
In an interview with Fox News's Jeanine Pirro on Friday, the president complained about hostility and unfair treatment at the hands of the media, suggesting for the second time in one day that he might just cancel the daily briefing - and adding that he might instead hold biweekly briefings himself. Mark WarnerMark WarnerDemocrat Senator warns of poor "outcome" for past secret Oval Office tapes Warner: I will "absolutely" subpoena tapes between Comey and Trump if they exist Dem senator: Trump administration "afraid" of Russian Federation investigation MORE (D-Va.) told host Chris Wallace on "Fox News Sunday".
White House press secretary Sean Spicer said Friday that it's not always possible to provide entirely accurate information at press briefings given President Trump's busy schedule. Senate Majority Leader Mitch McConnell (R-Ky.), Senate Minority Leader Charles E. Schumer (D-N.Y.) and House Speaker Paul D. Ryan (R-Wis.) declined, through representatives, to describe their conversations with Trump.
White House spokeswoman Sarah Huckabee Sanders disputed the report and said the president would "never even suggest the expectation of personal loyalty".
Trump also hit back on Friday at media reports questioning the credibility of White House accounts of why Comey was sacked and threatened an end to regular White House press briefings. "The best replacement for Comey is Trey Gowdy", said Bakari Sellers, who ran for SC lieutenant governor in 2014.
Fellow Republican Lindsay Graham of SC, speaking on NBC's "Meet the Press", said Trump now "has a duty and obligation to pick somebody beyond reproach outside the political lane" to lead the Federal Bureau of Investigation. "Because I can't trust that what I say to the president isn't going to be misconstrued and used against me because I don't do what he wants".
The evolving explanations from Mr. Trump and his team raise questions about why Rosenstein wrote the memo, CBS News chief Washington correspondent and anchor of "Face The Nation" John Dickerson said on the "CBS Evening News" Friday. Moscow has denied any such meddling.
CNN cited an unnamed source as saying Mr Comey was "not anxious about any tapes" Mr Trump may have of their conversations.
'There is no way to get at the story through talking to the White House press secretary or listening to the spin that is offered by officials who are willing to go on the record, ' says media critic Margaret Sullivan.