Comey, 50, told his top law enforcement agents from around the country that he had been asked by President Trump to stay on the job, according to people familiar with the matter.
A decision to retain Comey would spare the president another potentially bruising confirmation battle. It would also keep Comey at the center of the FBI's investigation into several Trump associates and their potential ties with the Russian government, The New York Times reported.
Comey's unusually public role in the election-season investigation into Clinton's handling of classified information on her private email server as secretary of state has made him a target of criticism from Democrats, many of whom believe he cost her the election.
The Justice Department's internal watchdog announced earlier this month an investigation into the department and the FBI's handling of the Clinton email probe.
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"He's become more famous than me," CNN quoted Trump as saying.
During the campaign, Trump had harshly criticised the FBI and the Justice Department for not bringing criminal charges against Clinton in connection with her use of a personal email server.
After Trump was elected in November, he said in a nationally televised interview that he had not made up his mind about whether he would ask Comey to resign.
Under federal law, the FBI director is appointed to a 10-year term, intended to overlap more than one administration as part of post-Watergate overhauls created to give the director independence and insulate the job from politics.
Comey will have to manage an increasingly difficult relationship with Trump and his White House, as the FBI is leading an investigation into ties between Trump's associates - including his former campaign manager, Paul Manafort - and the Russian government, the NYT said.
As part of that inquiry, agents have examined intercepted communications and financial transactions. Comey has repeatedly declined to discuss the investigation with members of Congress.
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