Mr Trump said it was "possible" he could name someone by Friday, before he leaves for his first foreign trip.
Comey's dismissal came on Tuesday, in what the White House alleged was a decision following the advice from the attorney general and deputy attorney general, although Trump later said he was intending to fire Comey "regardless". He said most are "very well known" and of the "highest level".
White House spokeswoman Sarah Huckabee Sanders disputed the report and said the president would "never even suggest the expectation of personal loyalty".
Trump said Saturday a new Federal Bureau of Investigation director could possibly be named in the next few days, but whoever it might be is likely to undergo intense questioning at a Senate confirmation hearing, delaying the time when the appointee might actually take over control of the agency.
A recent survey by the Pew Research Centre marking Trump's first 100 days in office, which was marked on April 29, found that just 39% of the general public approve of the new president.
Media captionWhat do Trump supporters think about Comey's firing?
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During the interview, Trump claimed that he fired Comey because he was a show-off.
President Donald Trump threatened Friday to "cancel all future "press briefings", saying it was impossible for his representatives to be perfectly accurate.
The FBI and at least two congressional committees are probing allegations of Russian interference into the 2016 USA presidential election and possible collusion by the Trump campaign team.
In the letter firing James Comey, President Trump "thanked" the former FBI Director for assuring him - "three times" - that he was not under investigation. Feinstein had already joined forces with the committee's chairman, Chuck Grassley of Iowa, to call for a briefing on the Russian Federation investigation.
As has happened on previous occasions since Trump took office, different versions rapidly circulated of an event, in this case phone conversations between Comey and Trump and a dinner the pair had at the White House.
During a Thursday interview with NBC's Lester Holt, Trump said he made the decision to fire Comey on his own, and that he would have done it regardless of what Rosenstein recommended, adding that he had been thinking of "this Russian Federation thing with Trump" when he came to his decision to fire Comey.