Lawmakers have not made any request for Ford to testify publicly before the Judiciary Committee.
"If Ms. Ford wishes to provide information to the committee, I would gladly listen to what she has to say and compare that against all other information we have received about Judge Kavanaugh", Graham said in a statement.
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'm urging Senate Republicans on the Judiciary Committee to delay a vote on Judge Kavanaugh", said Sen.
"I have never done anything like what the accuser describes - to her or to anyone", Kavanaugh said.
I agree with the concerns expressed in the Judiciary Committee's statement about the substance and process regarding the allegations in this latest claim against Judge Kavanaugh. She said she thinking that, "I'm not ever telling anyone this".
Then Ford came forward. "He was trying to attack me and remove my clothing".
That panel has a vote now set for Thursday afternoon on the Kavanaugh nomination. After initially guarding her anonymity, Ford "decided to take control of this and tell this in her own voice" after the allegations were leaked, her lawyer said.
Kavanaugh's new statement comes after the White House said the committee was leaning toward inviting his accuser, Christine Ford, to testify regarding the claims.
Feinstein issued a statement on Sunday afternoon in support of Ford and calling for the FBI to investigate the matter before the Senate advances Kavanaugh's nomination.
Feinstein said Sunday that the FBI should investigate the allegations against Kavanaugh before the committee moves forward with his nomination.
Since Wednesday, she has watched as that bare-bones version of her story became public without her name or her consent, drawing a blanket denial from Kavanaugh and roiling a nomination that just days ago seemed all but certain to succeed.
On July 30, Ford sent a letter about the allegation to Sen. "I did not do this back in high school or at any time".
"SenateGOP Male GOP Senators, huge opportunity here to pretend you give a shit about women & be the first to say you'll vote No on Kavanaugh".
"Unfortunately, committee Republicans have only known this person's identity from news reports for less than 24 hours and known about her allegations for less than a week", Grassley said.
Sixty-five women who knew Mr Kavanagh in high school defended him in another letter, circulated by Senate Judiciary Committee Republicans, as someone who "always treated women with decency and respect".
She alleged in an interview that Kavanaugh pinned her to a bed, clumsily tried to remove her clothing and covered her mouth when she tried to scream. Ford was 15 at the time of the alleged incident and Kavanaugh was 17. Grassley's office issued a statement that called Ford's account "uncorroborated allegations" and noted that no such accusations had turned up during the six times the Federal Bureau of Investigation has looked into Kavanaugh's background over his decades of public service.
Ford, a professor in clinical psychology at Palo Alto University, reluctantly came forward to tell her story, ultimately changing her life forever.
Ford said she told no one about the incident, until 2012.
One Judiciary Committee Republican, Jeff Flake of Arizona, told The Washington Post the accuser, Christine Blasey Ford, "must be heard" before the panel's vote on the nomination, now scheduled for Thursday.
Kavanaugh has denied the allegations, which have roiled his confirmation process. While it would be unusual, the committee is not required, however, to approve Kavanaugh before the full Senate votes.