Sen. Susan Collins of ME - one of three Republican senators who was instrumental last week in holding up Kavanaugh's confirmation vote - said she had extensive conversation with the White House counsel's office and is "confident that the FBI is doing a thorough investigation and that it will be helpful to us as we make our decisions".
Kavanaugh's confirmation hinges on a handful of key Republican and Democratic senators who have not yet fully tipped their votes.
In a statement released Sunday, a Yale University classmate of Brett Kavanaugh's said he is "deeply troubled by what has been a blatant mischaracterization by Brett himself of his drinking at Yale". "We have to allow them to go ahead".
Democrats expressed concern on Sunday about reported efforts to stymie the probe, which comes after the Judiciary Committee approved Mr Kavanaugh's nomination on Friday before it goes to the full Senate for a final vote.
He also directly criticized Trump, who fired him from his position at the bureau in 2017, writing: "We live in a world where the president is an accused serial abuser of women, who was caught on tape bragging about his ability to assault women and now likens the accusations against his nominee to the many "false" accusations against him". "I don't think this one will either", he said.
White House counsellor Kellyanne Conway said the investigation will be "limited in scope" and "will not be a fishing expedition". They forced the White House to open a supplemental background investigation of sexual misconduct allegations against the judge.
But with Republican leadership eager to place another conservative on the Supreme Court, it would be extremely unlikely that they would cooperate with a further delay of confirmation.
The person familiar with Ms Ramirez's questioning, who spoke on condition of anonymity, said she also provided investigators with the names of others who she said could corroborate her account.
Judge has written several books, including one titled "Wasted: Tales of a Gen X Drunk", and one titled "God and Man at Georgetown Prep: How I Became a Catholic Despite 20 Years of Catholic Schooling".
Trump said the charges of sexual assault against Judge Kavanaugh were made 36 years ago.
At a rally that drew hundreds to Boston City Hall, New York congressional candidate Alexandra Ocasio-Cortez warned Flake and other elected leaders that voters would end their careers should they support Trump's Supreme Court nominee. But as of Monday, Kavanaugh lacked the simple majority he needs for confirmation.
Deborah Ramirez has accused Kavanaugh of exposing himself to her at a party while they were both students at Yale.
Speaking during the Forbes Under 30 Summit, Flake said "any current, credible allegation that has been made" should be "fully investigated" during the FBI's weeklong probe that he called for on Friday.
Mr Kavanaugh denies the allegation. Mr Trump's fellow Republicans control the Senate by a narrow 51-49 margin.
Although there were wide gaps by gender, race and party affiliation, overall 48 percent of voters said that after last week's hearing featuring Kavanaugh and one of his accusers, Christine Blasey Ford, the Senate should give him the thumbs down, the Quinnipiac University survey found.
Mitchell also points out Ford claimed to want to keep her story confidential, but called the tipline at the Washington Post even before her claim was presented to Sen.
"FBI's hands must not be tied in this investigation".