The candidates are being looked at, White House officials said, with a particular emphasis in mind: a lack of deep ties to Mr. Trump, to avoid the appearance that he wants to install a crony at the top of an agency that is investigating the activities of his presidential campaign.
Attorney General Jeff Sessions and Deputy Attorney General Rod J. Rosenstein will conduct the interviews.
Here are some of the choices being discussed inside the White House:
Senator John Cornyn
Mr. Cornyn served as the Texas attorney general before being elected to the Senate in 2002, where he is the No. 2 Republican in the chamber. He is one of several possible candidates who are current and former members of the House and Senate.
But an adviser to Mr. Cornyn said the senator made clear in a conversation with the president on Wednesday that, while open to discussing the post, he is happy in the Senate. The adviser was not authorized to discuss the internal discussions and spoke on the condition of anonymity.
Andrew G. McCabe
Mr. Trump could also do something no president has: Elevate a senior F.B.I. official to be the next director. That would be seen as an olive branch to the bureau and would help avoid allegations that Mr. Trump was selecting a political loyalist.
Mr. McCabe, 49, is a career F.B.I. agent who was Mr. Comey’s deputy before moving in as acting director after Mr. Comey’s firing. He has been at the center of a number of high-profile counterterrorism investigations, including the one into the 2013 Boston Marathon bombing and the arrest and interrogation of a suspect in the 2012 attacks on the American compounds in Benghazi, Libya.
Judge Michael J. Garcia
Mr. Garcia spent nearly a decade as a federal prosecutor in New York City before President George W. Bush appointed him assistant secretary for Immigration and Customs Enforcement in 2003 and United States attorney two years later.
Before Gov. Andrew M. Cuomo appointed Mr. Garcia to New York State’s highest court in 2016, he worked in private practice and led a more than yearlong inquiry into FIFA, the global governing body of soccer that became the centerpiece of a sprawling corruption case announced by the United States in 2015. Mr. Garcia abruptly resigned from the investigation in 2014, disagreeing with the way the organization had characterized the results of his inquiry, which were never made public.
Ms. Fisher works as a white-collar criminal lawyer in Washington, and she ran the criminal division of the Justice Department during the second half of Mr. Bush’s administration.
J. Michael Luttig
Mr. Luttig is a former Justice Department lawyer and federal appeals court judge who was appointed by the first President George Bush. Widely admired by conservatives, Mr. Luttig left the bench in 2006 to become general counsel of Boeing, a position he still holds.
Representative Trey Gowdy
Mr. Gowdy, the South Carolina Republican and a former federal prosecutor, oversaw the Benghazi special investigation in the House. Many F.B.I. agents saw that inquiry as a boondoggle and view Mr. Gowdy as highly partisan.
Mr. Rogers is the former chairman of the powerful House Intelligence Committee, who represented Michigan and once served as an agent in the bureau. Respected among agents who felt he was a straight shooter, he probably has the credibility to steer the F.B.I. out of a hurricane of bad publicity.
Mr. Trump and his advisers have asked some people whether they believe Mr. Kelly, the former New York police commissioner, would be “loyal,” language similar to what he used in questioning Mr. Comey’s effectiveness, according to people briefed on the discussions. In Mr. Kelly’s case, they said, the White House appeared to be trying to assess whether he would seek to advance his own image, as Mr. Trump accused Mr. Comey of doing.
Mr. Kelly declined to comment on whether he had spoken with the president about the job. He is said to be seen as outside the top tier of candidates because the job is a 10-year appointment and at 75, he is older than many of the other prospects.
As police commissioner, he also had a combative relationship with the F.B.I., with the two agencies frequently engaged in turf wars. He was widely disliked among agents who felt he constantly worked to undermine the F.B.I., creating a legacy of mistrust.
Kelly Ayotte, the former New Hampshire attorney general who lost her seat in the United States Senate last year, has been pushed by some of her Senate colleagues and some Republican advisers to the White House who believe that she would easily be confirmed.
But Mr. Trump has remained leery of Ms. Ayotte, who publicly broke with him during his presidential campaign and criticized him, according to people close to the White House.
George Terwilliger, who helped lead the George W. Bush campaign’s recount efforts in 2000 and served as deputy attorney general under Mr. Bush’s father, is another name being batted about.
Chief among the White House’s concerns is that the Senate confirmation process go smoothly. Mr. Trump and his son-in-law and senior adviser, Jared Kushner, have made calls to people outside the White House asking their opinions of different candidates.