Trump repeatedly tried, without success, to get Federal Bureau of Investigation director James Comey to drop the probe into Flynn.
While the White House denied access to the meeting to USA media, the Russians published photos of Trump smiling and shaking hands with his Russian guests, a move that angered the White House.
Flynn was forced to resign one day before the conversation between Comey and Trump, after it was revealed that Flynn lied to Vice President Mike Pence about connections he had with Russian ambassador Sergey Kislyak. "I do (have confidence in Trump)", he told reporters at a news conference.
They asked the White House to turn over 'records of interactions with former Director Comey, including any audio recordings'.
Democratic Senator Mark Warren told United States media he believed "the American people will get a chance to hear from Comey shortly".
Members of the Senate Judiciary Committee also planned to ask Comey to testify, The Hill reported.
The sources for the Yahoo News story did not clarify how Trump contacted Flynn, and the White House did not immediately respond to a request for comment.
He added: "Look at the way I've been treated lately, especially by the media".
"You can't let them get you down". I guess that's why we won. Adversity makes you stronger. Don't give in, don't back down. "And the more righteous your fight, the most opposition you will face".
The White House has denied reports the embattled President had personally appealed to Comey to abandon the FBI's investigation into former National Security Adviser Michael Flynn's links to Russian Federation.
Twitter goes berserk after Donald Trump Jr. appears to confirm Comey memo
The Kentucky Republican had said initially that investigations by the Senate intelligence panel and the FBI were sufficient. Speaker because there is a belief in this country that no one is above the law and that includes the president".
The inspector general at the U.S. Defense Department announced recently it is investigating whether Flynn sought permission for the Russian and Turkish payments after he was specifically warned when he retired from the military to not accept any money from foreign governments.
The Comey firing and the reports of Trump's attempt to curb the Flynn investigation led to calls from lawmakers, Democrats and some Republicans, for appointment a special prosecutor in the Russian Federation investigation.
House and Senate committees have immediately - and rightly - requested copies of the memos related to this controversy.
Mitch McConnell, the Republican leader of the Senate, said simply, "It would be helpful to have less drama emanating from the White House".
The person who described the Comey memo to the AP was not authorised to discuss it by name and spoke on condition of anonymity.
The day after Flynn's dismissal, Tump implored former FBI Director James Comey to drop the FBI's investigation in Flynn, according to memos written by Comey immediately following the meeting.
The White House vigorously denied it all.
Democrats are pressing the Director of National Intelligence Dan Coates to assess the impact of President Trump sharing highly classified information with Russian officials during an Oval Office meeting last week.
The intensifying drama comes as Trump is set to embark Friday on his first foreign trip, which had been optimistically viewed by some aides as an opportunity to reset an administration floundering under an inexperienced president. Mark Warner, D-Virginia, sent a letter Wednesday asking Comey to testify before the committee in both open and closed sessions. "Memos, transcripts, tapes - the list keeps getting longer", he said.
The Congressman said that his call for impeachment is a position of conscience for him and asked the American people to speak up.