"There's clearly a lot of politics being played", House Speaker Paul Ryan said.
King, an independent who caucuses with Democrats, answered a question from CNN's Wolf Blitzer on whether the country "might be getting closer and closer to the possibility of the beginning of yet another impeachment process" if The New York Times' report on James Comey is true.
The associate who described the memo said Comey is willing to testify but wants to do it in public to ensure a full airing of events.
Trump himself has not weighed in on the story to confirm or reverse the White House's statement, as he has with this week's revelation that he revealed sensitive intelligence to Russian diplomats.
"We're not on anybody's side, ever", he said in a March speech.
At the top of the list were reports that President Trump had asked former FBI Director James B. Comey to drop an investigation into Michael Flynn, the former national security adviser. The White House disputed the account.
The memo appeared to show Mr Trump "trying to impede the investigation" into alleged Russian Federation links, according to David Gergen, who served as a top adviser to Presidents Bill Clinton, Ronald Reagan and Richard Nixon. Several members of Congress said that if Trump did suggest that Comey "let this go" regarding Flynn's Russian contacts, it was probably just a joke, light banter.
Mitch McConnell, the Republican leader of the Senate, said simply, "It would be helpful to have less drama emanating from the White House". I guess that's why we won.
-The Senate intelligence committee invited Comey to appear in both open and closed sessions. "And the more righteous your fight, the more opposition that you will face".More news: Mourinho: Why I Lost Temper With Wenger On Touchline
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The existence of Trump's request is the clearest evidence the president has tried to directly influence the Justice Department and FBI investigation into links between Trump's associates and Russian Federation, the newspaper reported. The Flynn investigation was part of a broader probe into Russian interference in last year's presidential election.
Armed Services Committee Chairman John McCain said late Tuesday that the developments had reached "a point where it's of Watergate size and scale".
Rep. Jason Chaffetz, Republican chairman of the House oversight committee, sent a letter to the Federal Bureau of Investigation on Tuesday requesting that it turn over all documents and recordings that detail communications between Comey and Trump.
Comey's memo, an apparent effort to create a paper trail of his contacts with the White House, would be the clearest evidence to date that the president has tried to influence the investigation.
FILE - In this February 1, 2017 file photo, National Security Adviser Michael Flynn speaks during the daily news briefing at the White House, in Washington.
The White House issued a denial of the Times report on Tuesday, saying the president "has never asked Mr. Comey or anyone else to end an investigation, including any investigation involving General Flynn".
Trump is preparing to leave town Friday on his first foreign trip, and aides have been hopeful the journey will be a chance for the administration to get back on track after weeks of chaos and distractions. "Reluctantly, I have to say yes", Sen.
Associated Press writers Vivian Salama and Jill Colvin contributed.