The Trump government - embroiled in a controversy that escalated with its sudden resignation and its different explanations of its reasons - will interview the top four candidates for the job on Saturday, the American media said.
"We can make a quick decision," Trump told reporters aboard the Air Force One presidential aircraft before flying to southern Virginia, where he plans to deliver a speech at a graduation from Liberty University, a center for evangelical study.
The White House has not set a deadline for the process of replacing Comey.
Asked if the decision could be made before his trip to Saudi Arabia on Friday, Trump replied, "even this is possible."
Trump described the candidates who are considered for the post as "prominent people," "very well-known," and "of the highest level."
On Saturday, Attorney General Jeff Sessions and his deputy Rod Rosenstein will meet with interim FBI Director Andrew McCabe, Texas Senator John Cornyn, former Attorney General Michael Garcia, and former Assistant Attorney General Alice Fisher, The New York Times quoted sources close to the meetings as saying.
There are almost a dozen candidates considered for the post.
The designation of a new FBI director who is independent of the White House will be closely scrutinized as Trump faces an avalanche of criticism for firing Comey, who was charged with a criminal investigation into the possible ties of his campaign team with Russia.
McCabe, a former FBI agent, has participated in many high-profile investigations, including the Boston marathon bombings in 2013 and the American consulate attack in Benghazi, Libya, in 2012.
This week McCabe contradicted Trump by telling Senators that Comey had strong support within the FBI and that the case of alleged Russian interference in the US election was a "highly significant investigation."
Cornyn, the Senate's Republican Party number two, was Texas attorney general before being elected senator in 2002.
Garcia was the number two Immigration and Customs Enforcement and Attorney General under former President George W. Bush.
Fisher led the criminal division of the attorney general's office under Bush.
Other candidates include former New York police chief Raymond Kelly, former Republican Senator Kelly Ayotte, and South Carolina lawmaker Trey Gowdy, a former attorney general who led the investigation into the Benghazi attack on the House of Representatives.
The one designated by Trump must be confirmed by the Senate, where Democrats and some Republicans have fiercely criticized Comey's dismissal.