Ford said she had kept silent about the alleged incident until she and her husband were in couples therapy in 2012.
Grassley made no mention of a public hearing for either party to address the allegation.
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.
"The group has also been promoting the hashtag "#IBelieveChristine" on it's Twitter account in an effort to boost pressure on GOP lawmakers.
Bloomberg News' Laura Litvan reports that Hatch was then asked how he would feel about Kavanaugh's nomination if it turned out that the allegations were true, and Hatch replied that "if that were true, I think it would be hard for senators not to consider who he is today".
The Post said Ford originally contacted her representative in the House, California Democratic Rep. Anna Eshoo, whose office sent her letter to Feinstein.
Kavanaugh was seen arriving at the White House, with no immediate reason given, while all 10 Democrats on the Senate Judiciary Committee wrote to Grassley asking him to postpone a scheduled Thursday vote on the nominee to give the Federal Bureau of Investigation more time to investigate.
New York Sen. Chuck Schumer, the chamber's top Democrat, called for Grassley to postpone the vote "until, at a very minimum, these serious and credible allegations are thoroughly investigated". In the wake of the explosive accusation, three Senate Republicans-senators Jeff Flake, Bob Corker and Lisa Murkowski -and a slew of Democrats have said the Senate Judiciary Committee should delay the vote. Kavanaugh has responded in a statement Monday by saying he is also willing to testify under oath. "I mean, I don't know her".
Mr Kavanaugh, 53, has released a statement calling the allegation against him "completely false".
Sen. Chuck Grassley, the chairman of the panel, said in a statement he was committed to hearing from Ford in an "an appropriate, precedented and respectful manner". She told reporters on Monday that she assumed committee staff interviews with Kavanaugh and Ford "would be the prelude to some sort of hearing".
Senate Judiciary Committee Chairman Chuck Grassley plans to speak with Kavanaugh and Ford before the committee's scheduled vote, according to a spokesman for the senator.
"I thought he might inadvertently kill me", said Ford, 51, a clinical psychology professor at Palo Alto University in California.
During his hearings allegations from a former colleague, law school professor Anita Hill, surfaced accusing him of repeated sexual harassment when they worked together. Katz said, "I would say no one in their right mind regardless of their motive would want to inject themselves into this process and face the kind of annihilation that she will be subjected to by those who want this nominee to go through". A positive vote would set up a debate following by a vote in the full Senate.
Senate Republicans are obviously disturbed, so they want to give Ford a chance to submit her accusations to the Senate to get to the truth of the matter. Susan Collins, a key Republican swing vote who has been at the center of the left's efforts to block the judge's nomination to the Supreme Court. "She was weighing her desire and her belief that she had a civic duty to provide this information to those making the decision about Brett Kavanaugh with, frankly, her fear about coming forward, and there was going to be great personal risk to her and her family in doing so".