Christine Blasey Ford has accused Mr Kavanaugh of trying to attack her and remove her clothing in the early 1980s when they were both high school students.
Debra S. Katz, the attorney for the accuser, said Ford was willing to tell her story publicly to the Judiciary panel but no lawmakers had yet contacted her. Katz denied that Ford, a Democrat, was politically motivated.
Some Republicans on the panel have said the accuser should have a chance to tell her story, a view echoed by White House senior adviser Kellyanne Conway.
Chuck Grassley (R-Iowa), the chairman of the Senate Judiciary Committee, is working to set up separate phone calls with Kavanaugh and Ford before the committee votes Thursday to advance Kavanaugh's confirmation to the Senate floor.
"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", Feinstein said.
From Kavanaugh: "I categorically and unequivocally deny this allegation", Kavanaugh said in a statement issued by the White House. She said if anyone was going to tell her story, she wanted to be the one to tell it.
Hill said in a statement to Politico, "The reluctance of someone to come forward demonstrates that even in the #MeToo era, it remains incredibly hard to report harassment, abuse or assault by people in power".
"My biggest fear was, do I look like someone just attacked me?" she said.
The Judiciary Committee has completed its hearings on Kavanaugh and plans to vote on Thursday on his nomination.
At the time, Ford said, she knew Kavanaugh and Judge as "friendly acquaintances" in the private-school social circles of suburban Maryland. Both loudly stumbled down the stair well at which point other persons at the house were talking with them. This is very significant for a man of character and integrity to be spoken about so highly by women who maybe didn't vote for President Trump, who maybe don't call themselves Republicans, certainly, in fact, many who don't.
Feinstein said Sunday that the FBI should investigate the allegations against Kavanaugh before the committee moves forward with his nomination. Another therapy session the following year includes the charge that Ford underwent a "rape attempt" in "her late teens", but she was allegedly 15 - not late teens - when this incident occurred.
CNN on Sunday published the contents of the letter Christine Blasey Ford wrote to Sen. "I did not do this back in high school or at any time", he said. She said she often spent time in the summer at the Columbia Country Club pool in Chevy Chase, where in those pre-cellphone days, teenagers learned about gatherings via word of mouth.
In the interview, Ford says she never revealed what had happened to her until 2012, when she and her husband sought couples therapy. Kavanaugh would have been 17 at the end of his junior year at Georgetown Prep.
The Post identified Ford, a 51-year-old research psychologist and professor at Palo Alto University in California, as the woman whose accusation surfaced last week and raised the first serious doubts about Kavanaugh's confirmation.
"It is quite possible-or even likely-that hearings won't prevent Brett Kavanaugh from being confirmed given the equivocal evidence against him and, perhaps even more important, the number of Republicans and red-state Democrats in the Senate".
Kavanaugh, now 53 and a federal appeals judge in Washington, has denied the allegation.
The FBI received a version of the letter with Ford's name redacted, according to a Republican official with knowledge of the letter, and then sent it to the White House to be included in Kavanaugh's background file.