Christine Ford, the California professor accusing Supreme Court nominee Brett Kavanaugh of sexual assault, told the Senate Judiciary Committee on Thursday that she “would be prepared to testify next week" -- apparently backing off her bid for the FBI to first launch a new inquiry.

According to an email sent by her attorney Debra Katz and first obtained by The New York Times, Ford would appear as long as senators provide “terms that are fair and which ensure her safety." Fox News has also obtained the email.

“As you are aware, she has 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,” the email continued. “She wishes to testify, provided that we can agree on terms that are fair and which ensure her safety.”

Senate Republicans had invited Ford to testify at a hearing on Monday, and gave her a Friday deadline to indicate whether she would attend.

A Monday hearing still appears unlikely, though, even though Kavanaugh said through the White House late Thursday that he'd be willing and ready to testify then.

In the letter Thursday, Ford's attorney wrote that it “is not possible" for Ford to testify on Monday. She added that "the Committee’s insistence that it occur then is arbitrary in any event.”

Ford's openness to testifying threatened to once again upend Kavanaugh's confirmation process. Senate Republicans had said that if Ford stuck to her apparent refusal to testifiy, they would have moved forward with a vote on Kavanaugh on Wednesday. That vote is now in doubt.

However, the attorneys concluded: "Her strong preference continues to be for the Senate Judiciary Committee to allow for a full investigation prior to her testimony. "

Late Tuesday, Ford's lawyers strongly suggested she would testify only if the FBI first conducted a "full investigation."

INCONSISTENCIES EMERGE IN KAVANAUGH ACCUSATIONS AS WITNESS DELETES ONLINE ACCOUNT

Senate Judiciary Committee Chuck Grassley, R-Iowa, pushed back sharply on that request, writing in a letter that it was not the FBI's “role to investigate a matter such as this" and that the agency already had reviewed the allegations and supplemented its background check of Kavanaugh.

And in a series of tweets earlier Thursday, Republicans on the Senate Judiciary Committee noted they had obtained statements, under penalty of felony, from three people at the house party where the alleged assault occured, including Kavanaugh, his friend Mark Judge, and another individual. Fox News has learned that Kavanaugh provided his statements under oath, exposing him to potential perjury liability if he were lying.

Committee members also wrote that they had reached out to a "fourth person allegedly at the party," as well as "a schoolmate who claimed on social media this week to have info related to Dr. Ford’s allegations" -- but had not heard back.

That was an apparent reference to a widely circulated online account by Cristina Miranda King, who claimed that she heard about the alleged assault at the time. King deleted her online post after questions emerged about apparent inconsistencies.

"[Ford's] attorneys say there needs to be an investigation, which is exactly what the committee has been doing all week," the GOP members wrote. "And we would love to hear from Dr. Ford. Democratic staff is invited to participate fully every step of the way."

Meanwhile, tensions over Kavanaugh's nomination were evident Thursday on Capitol Hill. A total of 33 protesters were arrested outside Grassley's office in the Hart Senate Office Building this afternoon, and 23 were booked in the morning in the Dirksen Senate Office Building. They were charged with obstructing and crowding, according to Capitol Police officials.

Images posted on social media showed several demonstrators eating lunch in Grassley's office as aides took notes of their complaints.

WATCH: FEINSTEIN ADMITS SHE'S NOT SURE ACCUSER BEING ENTIRELY 'TRUTHFUL'

Sources tell Fox News that several death threats have been sent to Kavanaugh and his family, as well as Ford. Maine Republican Sen. Susan Collins, a moderate and potential key swing vote in Kavanaugh's confirmation, also said she had been receiving threatening messages.

In response to Collins' complaint, a top Democratic California lawmaker, Rep. Eric Swalwell, tweeted, "Boo Hoo Hoo." He later deleted the tweet and called it "stupid."

Late Thursday, Democrats on the Judiciary Committee wrote a letter demanding the FBI investigate threats against Ford.

"We are increasingly concerned by reports of alleged death threats against her, the hacking of her email, and the harassment and intimidation [Dr. Blasey Ford] has faced since being forced into the spotlight late last week," the Democrats wrote, adding that they "believe that how Dr. Blasey Ford is treated in this moment reflects upon how seriously our Nation treats credible claims of sexual assault, and whether we have learned from past mistakes."

Ford, who has acknowledged she could not remember exactly when or where the alleged sexual assault incident happened more than three decades ago, had been under intense pressure to make her allegations against Kavanaugh under oath. She reportedly had asked a friend over the summer whether she had ever confided in her that she had been sexually assaulted, and was told that she had not.

Kavanaugh denied the allegations when he spoke under oath with Senate Republican staff earlier his week, Fox News has learned. Democratic Judiciary Committee members and staff refused to participate in those closed-door proceedings.

The California professor said she was reluctant to come forward and did so only because her hand was forced by the media. She wrote a letter outlining her allegations in July to Sen. Dianne Feinstein, D-Calif., but Feinstein did not mention them to other senators or federal investigators until last week.

Feinstein, the top Democrat on the Judiciary Committee, has said she was motivated by a desire to protect Ford's identity, but Republicans countered that Feinstein could have mentioned the allegations without naming Ford.

According to a scathing letter sent by Grassley to top Democrats this week, Feinstein actually compromised Ford's privacy by waiting so long and allowing information about her letter to leak to the media just days before a key vote on Kavanaugh's confirmation.

Grassley also blasted Feinstein for failing to disclose to Republicans an unredacted version of the July 30 letter she received from Ford.

"I cannot overstate how disappointed I am," Grassley wrote, charging that Feinstein "chose to sit on the allegations until a politically opportune moment."

Fox News' Chad Pergram and Alex Pappas contributed to this report.