The Iowa Republican told reporters Wednesday he is focused in the next few days on "doing everything we can to make Dr. Ford comfortable with coming before our committee, either in an open session or a closed session, or a public or a private interview".
Kavanaugh will not be the first to testify as requested by Ford's attorneys; he will appear after Ford testifies.
A person familiar with the call said Ford could not testify until next Thursday.
Kavanaugh denies any attack occurred, and Republicans have appeared poised to seek a confirmation vote as early as next week if she does not agree to testify. It said the telephone discussion ended with no decision. This follows a week where Ford and her lawyers were locked in negotiation with Sen. CNN said the lawyers told the committee staff that she wanted to be questioned by senators.
"As you are aware, she's been receiving death threats which have been reported to the Federal Bureau of Investigation and she and her family have been forced out of their home", Katz wrote to the committee.
Kavanaugh is a District of Columbia appeals court judge and served as an adviser to former President George W Bush.
Kavanaugh denied the allegations when they initially surfaced anonymously after Ford detailed them in a letter to her local Congresswoman and then to California Senator Diane Feinstein.
Ford's lawyers said she wants the FBI to investigate her allegation that she was sexually assaulted by President Trump's SCOTUS pick before she testifies at a Senate Judiciary Committee hearing.
The threats were confirmed Thursday by a senior administration official at the White House who was not authorized to speak publicly and disclosed the news on condition of anonymity.
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Later Wednesday, Grassley wrote to the senior Democrat on the committee, Dianne Feinstein, requesting she provide an unredacted copy of the letter Ford sent her in July about the assault allegation, saying he must review it before Monday's hearing.
Ford's lawyers continue to say that they want a full FBI investigation to take place before Ford will testify. Bredesen said September 19 that the woman accusing Supreme Court nominee Brett Kavanaugh of sexual assault "has a very credible story".
A spokesman for the chairman of the Senate Judiciary Committee is confirming that the committee will hold a public hearing Thursday on the sexual assault allegation that's been made against Supreme Court nominee Brett Kavanaugh.
"Really, they're hurting somebody's life, " he said of the senators considering Kavanaugh's nomination.
Groups of protesters, majority women, clogged the lobby of Grassley's Senate office on Thursday and targeted other Republican lawmakers.
Even President Trump has shifted his tone on Ford's allegations over the last 48 hours.
Ford came forward with the allegation in an interview published in the Washington Post on Sunday. At least he did in 2015, alluding to secrecy at the high school he attended when he allegedly pinned down a 15-year-old girl and groped her.
A statement by a Ford attorney, Lisa Banks, says Grassley's plan to call just two witnesses, Kavanaugh and Ford, "is not a fair or good faith investigation".