Democrats immediately called for a delay in a key committee vote set for this later week and a Republican on the closely divided panel said he's "not comfortable" voting on the nomination without first hearing from the accuser.
The woman, Christine Blasey Ford, told The Washington Post in her first interview that Kavanaugh pinned her to a bed at a Maryland party they attended in the early 1980s, clumsily tried to remove her clothing and put his hand over her mouth when she tried to scream.
"I thought he might inadvertently kill me," Ford said.
"He was trying to attack me and remove my clothing." Ford, 51 and a clinical psychology professor at Palo Alto University in California, says she was able to get away after a friend of Kavanaugh's who was in the room jumped on top of them and everyone tumbled.
Kavanaugh, 53 and a federal appeals judge in Washington, on Sunday repeated an earlier denial of Ford's allegation.
"I categorically and unequivocally deny this allegation. I did not do this back in high school or at any time," Kavanaugh said through the White House.
The allegation first came to light late last week in the form of a letter that has been in the possession of Sen Dianne Feinstein of California, the top Democrat on the Senate Judiciary Committee, for some time.
The committee recently concluded four days of public hearings on the nomination and the panel's Republican chairman, Chuck Grassley of Iowa, scheduled a Thursday vote on whether to recommend that the full Senate confirm Kavanaugh for a lifetime appointment to the nation's highest court.
Democrats, led by New York Sen Chuck Schumer, immediately called for it to be postponed, though Republicans gave no indication Sunday that they would accede to the calls by Democrats, most of whom already publicly oppose Kavanaugh.
A spokesman for the Senate Judiciary Committee said late Sunday that Grassley is trying to arrange separate, follow-up calls with Kavanaugh and Ford, but just for aides to Grassley and Feinstein, before Thursday's scheduled vote.
Feinstein late Sunday rejected Grassley's plan for phone interviews with Kavanaugh and Ford.
"Staff calls aren't the appropriate way to handle this," Feinstein said.
Sen Jeff Flake, R-Ariz, a committee member, told The Washington Post and Politico in interviews Sunday that he's "not comfortable" voting for Kavanaugh until he learns more about the allegation.
Flake is one of 11 Republicans on the committee, whose 10 Democrats all oppose Kavanaugh.
A potential "no" vote from Flake would complicate Kavanaugh's prospects.
Feinstein applauded Flake's call to slow things down.
"I agree with Senator Flake that we should delay this week's vote," she said.
"There's a lot of information we don't know and the FBI should have the time it needs to investigate this new material."
Another Republican member, Sen Lindsey Graham of South Carolina, said he's willing to hear from Ford provided that it's "done immediately" to keep the confirmation process on track.
Critics have accused the GOP of fast-tracking the process to get Kavanaugh on the court by October 1, the first day of the fall term.
Senate Republicans, along with the White House, see no need to postpone voting over what they consider uncorroborated and unverifiable accusations, according to a person familiar with the situation but not authorised to speak publicly.
In considering their options Sunday, Republicans largely settled on the view that Ford's story alone was not enough to delay Kavanaugh's confirmation.
Grassley could invite Ford to testify, likely in closed session before Thursday. Kavanaugh would also probably be asked to appear before senators.
The panel would also likely seek testimony from Mark Judge, Kavanaugh's friend and classmate who Ford says jumped on top of her and Kavanaugh.
Judge has denied that the incident happened.
Republicans have not settled on the strategy, the person familiar with the situation said, but were weighing options, including doing nothing.
Republicans say the allegations have already cast a shadow over Kavanaugh but that it does not appear to be enough to change the votes in the narrowly divided 51-49 Senate.
Key will be the views of Sens Susan Collins of Maine and Lisa Murkowski of Alaska, who did not immediately comment publicly Sunday.
A spokesman for Grassley said Kavanaugh already went through several days of hearings and has been investigated by the FBI.
The White House has accused Feinstein, who revealed the letter's existence late last week, of mounting an "11th hour attempt to delay his confirmation."
The White House has also sought to cast doubt about Ford's allegation by noting that the FBI has repeatedly investigated Kavanaugh since the 1990s for highly sensitive positions he has held, including in the office of independent counsel Ken Starr, at the White House and his current post on the federal appeals court in Washington.
Both Democratic and Republican senators questioned Feinstein's handling of the allegation.
Kavanaugh's nomination has sharply divided an already closely divided Senate, with most Democrats opposing him and most Republicans supporting him.