The leaders of the House Russia investigation, Michael Conaway, a Texas Republican, and Adam Schiff, a California Democrat, sent a request directly to Comey for copies of his memos, CNN reported.
They sent a separate request to Trump's White House counsel, Don McGahn, for any record the White House had of their meetings.
The House requested Comey and the White House provide the records, including tapes if they exist, by June 23.
Richman confirmed Thursday that he was the friend Comey testified who had discussed the memos with the press.
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Richman has been in touch with Senate judiciary through special counsel Bob Mueller's office, and that there will be movement after the weekend, a source was quoted as saying.
"The matter will be addressed on Monday," the source said.
It was unclear if the memos will be turned over to Congress on Monday, the report said.
Four members of the Senate Judiciary Committee, including Chairman Chuck Grassley and ranking Democrat Dianne Feinstein, had set a Friday deadline for Richman to respond after Comey testified that he gave the memo to him and asked him to anonymously disclose it to the media in an attempt to prompt the naming of a special counsel.
Comey testified Thursday that Trump asked him to drop an investigation into former national security adviser Michael Flynn and implied Trump may have become the subject of investigation himself for interfering in the FBI's work.
Trump said he is "100 per cent" willing to testify under oath on Comey meetings and tell the FBI Special Consul Robert Muller that he never asked FBI to stop investigating.
Trump also denied asking Comey for a pledge of loyalty.
Comey repeatedly called Trump a liar -- Trump responded Friday by saying the sacked FBI director was himself a liar.
Asked if he had taped their conversations, Trump said he would reveal that answer soon.
During the news conference, Trump said he would have more information related to whether tapes exist at a later date.
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