It comes as Mr Trump earlier said he wants the FBI to conduct a comprehensive and quick investigation into sexual misconduct allegations against his US Supreme Court nominee Brett Kavanaugh, but said he did not want to see a "witch hunt".
"I hope that someone does run in the Republican primary, somebody to challenge the president", Flake said. But he said the statements Kavanaugh is making now do matter.
Despite an emotional testimony from Christine Blasey Ford, who accuses Kavanaugh of sexually assaulting her as a high schooler, the committee voted Friday to advance the confirmation to the Senate floor for a final vote.
Trump on Monday said it "wouldn't bother" him at all if the Federal Bureau of Investigation questioned all three women who've accused Kavanaugh of sexual misconduct, but he also described Swetnick as having "very little credibility".
Although the Senate Judiciary Committee on Friday approved Judge Kavanaugh's nomination, it was conditional on a new inquiry into what it said were the "credible allegations" facing him.
President Donald Trump defended his Supreme Court nominee, Brett Kavanaugh, during a Monday press conference, calling the judge's Senate confirmation process "not fair". We could call it 'As the Stomach Turns.' If you think this is a search for the truth, you probably ought to put down the bong.
Democratic Senator Chris Coons told reporters he was in discussions with the White House on the probe. They forced the White House to open a supplemental background investigation of sexual misconduct allegations against the judge.
Senate Judiciary Chairman Chuck Grassley, an Iowa Republican, declined on Monday afternoon to provide more detail on what the FBI's investigation can cover. "At the same time, just so we all understand, this is our seventh investigation of a man who has really.you look at his life until this happened, what change he's gone through, what his family's gone through", he said.
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Chad Ludington, a Yale classmate of Kavanaugh, said in a media statement the judge was not truthful during his Senate testimony last week about his drinking and that Kavanaugh was "a frequent drinker and a heavy drinker" at Yale who often got belligerent and aggressive when drunk. The one question I didn't ask is how about the last 20 years?
GOP leaders reluctantly agreed to a limited investigation lasting no longer than one week due to their razor-thin majority in the Senate.
Supreme Court nominee Brett Kavanaugh testifies before the Senate Judiciary Committee on Capitol Hill in Washington, on September 27, 2018.
In his testimony, Kavanaugh vehemently denied Ford's claim.
Quoting an unnamed senior United States official, NBC reported that the FBI's witness list does not include Julie Swetnick, the third accuser to come forward with a claim against Mr Kavanaugh.
Despite this, NBC news quoted a White House official as saying restrictions remained in place, adding that as the FBI is carrying out a background - not a criminal - investigation, the White House decides the parameters of the probe. But as of Monday, Kavanaugh lacked the simple majority he needs for confirmation.
Republicans have 51 seats in the closely divided 100-member Senate and can not afford to lose more than one vote on confirmation.
The third woman, Julie Swetnick, accused Kavanaugh and Judge of excessive drinking and inappropriate treatment of women in the early 1980s, among other accusations.
"Unfortunately, despite having the time and opportunity to do so, Senator McConnell has refused to allow that FBI investigation".