The associate told Reuters that the details of the document as first reported by the New York Times were accurate.
"I hope you can let this go," Trump told Comey, according to the language of the memo, which the source confirmed.
The White House denied the report in a statement to reporters, saying it was "not a truthful or accurate portrayal of the conversation between the President and Mr. Comey."
The memo was written by Comey immediately after his meeting in the White House Oval Office with Trump one day after Flynn resigned over his contacts with Russians.
Flynn's resignation came hours after it was reported that the Justice Department had warned the White House weeks earlier that Flynn could be vulnerable to blackmail for contacts with Russian ambassador Sergei Kislyak before Trump took power on Jan. 20.
A spokeswoman for the FBI declined to comment on the details of the memo.
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