US Supreme Court nominee Brett Kavanaugh speaks on the second day of his confirmation hearing in front of the US Senate on Capitol Hill in Washington DC, on September 5, 2018.

Christine Blasey Ford, who alleged Supreme Court nominee Brett Kavanaugh sexually assaulted her when they were teenagers, told the Senate Judiciary Committee she is open to testify next week if they can "ensure her safety" and offer "terms that are fair."

Ford's email to the Senate Judiciary Committee also said that testifying on Monday, as committee Chairman Chuck Grassley, R-Iowa, had scheduled, "is not possible," adding that the "insistence that it occur then is arbitrary in any event."

The New York Times first reported the email on Thursday.

Ford's lawyer, Debra Katz, also said in the email that she would like to set up a call on Thursday to "discuss the conditions" required for Ford to testify.

"We are happy that Dr. Ford's attorneys are now engaging with the Committee," a spokeswoman for Grassley told NBC News later on Thursday.

Ford, 51, alleged in a letter obtained by Sen. Dianne Feinstein, D-Calif., in late July that Kavanaugh, 53, had pinned her to a bed, covered her mouth and attempted to take her clothes off while intoxicated at a gathering in the 1980s.

Kavanaugh, President Donald Trump's second Supreme Court nominee, strongly denies the allegations and has said he is willing to testify under oath in a public Senate hearing Monday.

In a letter to Grassley sent later on Thursday, Kavanaugh said he will attend the Monday hearing as scheduled.

"I continue to want a hearing as soon as possible, so that I can clear my name," he said in the letter.

Katz had signaled that Ford would be willing to testify on the same date, but later said that the FBI should complete an investigation into the allegations before Ford is willing to testify.

Grassley, in a letter to Ford's lawyers sent Wednesday, said that the lawyers must send her prepared testimony and biography documents to the committee by Friday morning if Ford was to be allowed to testify at the hearing on Monday.

Katz's email says that Ford "has been receiving death threats," and that "she and her family have been forced out of their home" after her identity was revealed over the weekend. Ford's initial letter to California Democratic Rep. Anna Eshoo requested she remain anonymous.

Kavanaugh's wife, Ashley Estes Kavanaugh, has also received threats of violence, a senior Trump administration official told The Wall Street Journal on Thursday. The U.S. Marshal Service is investigating the threats, which include two profane emails sent from the same address, the official told the Journal.

Some Republicans have balked at requests to complete an FBI investigation — which many Democrats have called for — suggesting that a probe is intended to delay Kavanaugh's nomination past the November midterm elections.

Politico reported Thursday that Ford was being advised by Ricki Seidman, a Democratic Party operative who had worked for the administrations of Barack Obama and Bill Clinton.

NBC News, citing a person involved in the confirmation process, reported that Kavanaugh was preparing for the hearing at the White House on Tuesday with coaching from senior officials including press secretary Sarah Huckabee Sanders and White House counsel Don McGahn.

Neither the White House nor the committee majority's office immediately responded to CNBC's requests for comment.