Mr Trump sacked Mr Comey last year while he was leading an FBI investigation into allegations of Russian interference in the 2016 presidential election.
Since then the men have been locked into a bitter and public war of words.
Mr Comey has painted Mr Trump as “morally unfit” to be US president, and suggested he may have obstructed justice by trying to pressure him into dropping any investigation into Mr Flynn.
Mr Flynn was forced to resign over charges that he had lied to the FBI about his contacts with Russia.
So what is in the memos?
Mr Comey’s partially redacted memos were handed to Congress on Thursday.
Details published in Mr Comey’s 15 pages of notes are consistent with allegations in his new book, A Higher Loyalty: Truth, Lies, and Leadership.
In his memos, Mr Comey writes that he was asked by Mr Trump to drop an inquiry into links between Mr Flynn and Russia.
“I hope you can let this go,” Mr Trump is reported to have said after a White House meeting. Mr Trump strongly denies Mr Comey’s account.