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In an interview with the Washington Post, Christine Blasey Ford said that as a high school student in suburban Maryland decades earlier, a "stumbling drunk" Kavanaugh pinned her to a bed, groped her and attempted to remove her clothing.
Last week, Kavanaugh, Republican President Donald Trump's second nominee for a lifetime appointment to the nation's highest court, said he "categorically and unequivocally" denies the allegations.
The White House did not immediately reply to a request for comment.
Senate Judiciary Committee spokesman Taylor Foy said Senator Chuck Grassley, the panel's Republican chairman, was working to set up follow-up calls with Kavanaugh and Ford before the committee's scheduled vote on Kavanaugh on Thursday, given the revelation of Ford's identity and "the late addendum to the background file."
"The Chairman and Ranking Member routinely hold bipartisan staff calls with nominees when updates are made to nominees' background files," Foy said in a statement.
Because Trump's fellow Republicans control a slim 51-49 majority in the Senate, Democrats cannot stop Kavanaugh's appointment unless some Republicans make a rare decision to break with their party and vote against Trump.
At least one Republican member of the Judiciary Committee, Senator Jeff Flake, told the Post on Sunday that Ford "must be heard" and urged the panel not to vote on Kavanaugh's nomination until it can hear from her
Republicans hold only an 11-10 majority on the committee, so Flake's vote could make a difference.
The Senate Judiciary Committee has completed its hearings on Kavanaugh and plans to vote on Thursday on his nomination. A positive vote would set up a debate following by a vote in the full Senate.
Two moderate Republican female senators who support abortion rights and are not on the Judiciary Committee, Susan Collins and Lisa Murkowski, have been under particular pressure not to back Kavanaugh, who is seen as a social conservative.
Responding to the latest developments in the Kavanaugh nomination, Foy had said, "It’s disturbing that these uncorroborated allegations from more than 35 years ago, during high school, would surface on the eve of a committee vote."
Senator Lindsey Graham, a senior Republican committee member, said he "would gladly" hear from Ford if she wanted to appear before the panel, but it would have to be done quickly.
"If the committee is to hear from Ms. Ford, it should be done immediately so the process can continue as scheduled," Graham said in a statement.
PRESSING FOR A DELAY
Democrats sought to put pressure on Republicans to delay the process.
Senate Democratic Leader Chuck Schumer said the Judiciary Committee "must postpone the vote until, at a very minimum, these serious and credible allegations are thoroughly investigated."
Senator Diane Feinstein, the top Judiciary Committee Democrat, issued a statement calling for a delay.
"I support Mrs. Ford’s decision to share her story and now that she has, it is in the hands of the FBI to conduct an investigation. This should happen before the Senate moves forward on this nominee," Feinstein said.
In the Washington Post interview, Ford said that when she tried to scream, Kavanaugh put his hand over her mouth. "I thought he might inadvertently kill me," Ford told the newspaper, adding, "He was trying to attack me and remove my clothing."
McConnell has said he would like to have Kavanaugh confirmed by Oct. 1, the start of the Supreme Court's new term.
That timetable would have Kavanaugh sitting on the Supreme Court - if he is confirmed - before Election Day on Nov. 6, when one-third of the 100 Senate seats are up for grabs.
Ford, now a 51-year-old research psychologist in California, told the Washington Post that in July she sent a letter to Democratic Representative Anna Eshoo about the incident but requested confidentiality at the time.
The existence of the letter and some details of its contents became public in recent days, however.
Ford did not immediately respond to a request from Reuters for comment.