Ford's lawyers said on Tuesday she would not appear before the Senate until after the FBI investigates the allegations. Mark Judge, the other person supposedly in the room during the incident, claims he never saw Kavanaugh behave how Ford describes.
Kavanaugh was at the White House for a second straight day on Tuesday, but again did not meet with Trump.
But Susan Collins, a Republican senator seen as potentially voting against party lines, also said that Ford needed to testify.
The public accusation led Grassley on Monday to announce that the committee would convene next Monday to give both Kavanaugh and Ford the opportunity to publicly testify, as both had indicated a willingness to do so.
Prof Ford's legal team has declined the Senate's offer to testify.
Why did the Senate set a deadline?She said she interacted with Kavanaugh hundreds of times while they were students in a close-knit constellation of single-sex Catholic schools around Washington in the 1980s. Republicans, however, hold a 51-49 majority in the chamber and could still confirm Kavanaugh's appointment even if all Democrats were to oppose his nomination.
But Republicans are hoping to win committee and full Senate approval for Mr Kavanaugh this month, before the new Supreme Court session begins, and have shown no taste for slowing that process.
Here's Jim Acosta asking Gillibrand one of his patented "tough questions" - what would you say to Ford if you could? - which is Gillibrand's cue to feign outrage and start babbling about how the Judiciary Committee hasn't caught up to the rest of the world 'n stuff.
"The FBI does not make a credibility assessment of any information it receives with respect to a nominee".
The White House under President George W Bush ordered that inquiry.
Ever heard of Anita Hill?
Judge Thomas was narrowly confirmed and he remains a Supreme Court justice.
Grassley added that his staff would be willing to speak to Ford "at a time and place convenient to her".
Without directly referencing Trump Jr., Conway said Ford "should not be insulted, and she should not be ignored".
"You have stated repeatedly that Dr Ford wants to tell her story". He has told advisers that he is skeptical of Ford's account and the timing of the allegation, these advisers said.
He called the allegation against Judge Kavanaugh "a drive-by shooting".
On Tuesday, a woman named Christina King said on Twitter that she was a former classmate of Kavanaugh's and had signed a letter in support of Ford, who has claimed she was sexually assaulted in high school.
She says he pinned her to a bed and clamped his hand over her mouth when she tried to scream.
Kavanaugh, Judge, and Smyth are saying that they don't remember a party "like" the one Ford described.
What are Trump and other Republicans saying?
"I really want to see her, to see what she has to say", Trump said.
McConnell said that "blatant malpractice" by Democrats - not releasing a letter by the accuser until the confirmation process was nearing its end - "will not stop the Senate from moving forward in a responsible manner". "We've invited everybody", he said. Ford and Judge Kavanaugh to testify under oath next Monday.
"Well, I would let the senators take their course", he said.