Comey informed senior agents at the Federal Bureau of Investigation about Trump's decision during a conference call last week, The New York Times reported, citing unnamed people familiar with the matter.
The FBI director is a Republican who was appointed by former president Barack Obama in 2013.
An FBI spokeswoman declined to comment when asked about The New York Times report. The Washington Post also reported on Comey's decision to stay on, citing unnamed sources.
Relations between the current White House and the FBI are especially sensitive because the Bureau is currently investigating potential ties of several Trump associates to Russian officials.
Trump told Comey during their first meeting at Trump Tower in New York earlier this month that he hoped he would remain in office, the Times reported, citing anonymous sources.
"And Mr Trump's aides have made it clear to Mr Comey that the president does not plan to ask him to leave," the paper added at the time.
Comey first angered Republicans in July by recommending that Clinton not be prosecuted Clinton, though he called her actions "extremely careless."
Eleven days before the November 8 election, he prompted more shock and dismay, this time among Democrats, by informing Congress that the FBI was reopening the inquiry into Clinton after some of her emails were discovered on the computer of an aide's estranged husband.
Two days before the vote, the FBI said the emails contained no new relevant information.
Clinton and many other Democrats blame Comey's 11th-hour revelation for her defeat.
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