Former FBI director James Comey is willing to testify before the Senate, but he will only do so in public, a close associate told The New York Times.
"We can make a fast decision", Mr Trump said aboard Air Force One before flying to southern Virginia, where he delivered a commencement address at Liberty University, an evangelical Christian school.
"Even that is possible", Trump said, when asked if it could be before he left.
The FBI has never had a female director.
One of those being interviewed is Republican Senator John Cornyn, the second highest-ranking member of the Senate and a former Texas Attorney General.
"They've been vetted over their lifetime essentially, but very well known, highly respected, really talented people. And that's what we want for the Federal Bureau of Investigation".
Barr assured investigations would continue, including the Clinton email scandal and claims of Russian involvement in USA elections, and the agency would "do a thorough and professional job regardless of who is serving as the bureau's director".
The White House said Mr Comey was sacked over the FBI's mishandling of the investigation into Hillary Clinton's email practices. Before that, he was the assistant secretary for Immigration and Customs Enforcement, appointed by President George W. Bush.
The Trump administration is looking to fill the job, which requires Senate confirmation, after Trump abruptly fired Director James Comey on Tuesday. "By unilaterally announcing his conclusions regarding how the matter should be resolved, Comey arrogated the attorney general's authority to himself", Barr, who served as attorney general from 1991 to 1993, wrote.
Other candidates include former NY police commissioner Raymond Kelly, former Republican senator Kelly Ayotte, and SC congressman Trey Gowdy, a former federal prosecutor who headed the Benghazi investigation in the House. Trump tweeted on Friday.
Mr. Trump will travel to Saudi Arabia at the end of next week, followed by visits to the Vatican and Israel.
The president kept to an upbeat message and did not mention the fallout over Mr Comey's firing in his remarks.
In an interview with U.S. media, President Trump also accused him being a "showboat" and a "grandstander".
"Following your convictions means you must be willing to face criticism from those who lack the same courage to do what is right, and they know what is right but they don't have the courage or the guts or the stamina to take it and to do it".
Mr Clapper said America's founding fathers had created three co-equal branches of government with checks and balances, but with Mr Trump as president, that was now under assault and "eroding".