Mr Comey is among officials who will remain in their positions under the new administration, according to a Justice Department memo.
Hillary Clinton blamed her election loss on the FBI reopening an inquiry into her emails 11 days before voting.
FBI directors serve 10-year terms and Mr Comey was appointed by former President Barack Obama in 2013.
The president's decision to keep the FBI director in his job was announced in a conference call by Mr Comey to the bureau's special agents last week, the New York Times reports.
The choice has yet to be officially confirmed by the Trump administration.
On Sunday at the White House, Mr Trump met Mr Comey at an event for law enforcement officials.
"Oh, there's Jim, he's become more famous than me," the president said before embracing the FBI chief.
The agency is currently investigating ties between Mr Trump's associates - including his former campaign manager, Paul Manafort - and the Russian government.
The Justice Department's inspector general, which oversees the FBI, is meanwhile investigating Mr Comey over his handling of the Clinton emails inquiry.
Republicans, including Mr Trump, had also criticised Mr Comey for failing to prosecute the Democratic candidate, who kept a private email server at her home.
The FBI director had called Mrs Clinton's handling of classified information "extremely careless".
Embattled FBI Director James Comey will stay in his current role, reportedly at the request of President Donald Trump.