(CNN) The woman accusing Supreme Court nominee Brett Kavanaugh of sexual assault says the FBI should investigate the incident before senators hold a hearing on the allegations.

In a letter addressed to Senate Judiciary Chairman Chuck Grassley of Iowa, and obtained by CNN's "Anderson Cooper 360," Christine Blasey Ford's attorneys argue that "a full investigation by law enforcement officials will ensure that the crucial facts and witnesses in this matter are assessed in a non-partisan manner, and that the Committee is fully informed before conducting any hearing or making any decisions."

The letter from Ford's lawyers notes that despite receiving a "stunning amount of support from her community," Ford has also "been the target of vicious harassment and even death threats" and has been forced to leave her home.

"We would welcome the opportunity to talk with you and Ranking Member Feinstein to discuss reasonable steps as to how Dr. Ford can cooperate while also taking care of her own health and security," the letter from Ford's lawyers said.

Ford's attorney Lisa Banks told Cooper that Ford will talk with the committee but added, "She is not prepared to talk with them at a hearing on Monday."

"She will talk with the committee," Banks said. "She is not prepared to talk with them at a hearing on Monday. This just came out 48 hours ago."

"Asking her to come forward in four or five days and sit before the Judiciary Committee on national TV is not a fair process. If they care about doing the right thing here and treating this seriously as they have said, then they will do the right thing and they will properly investigate this, and she will work with them in that investigation and also to share her story with the committee," Banks said Tuesday night.

Grassley said in a statement Tuesday night that there were never any plans to sit Ford and Kavanaugh together at the witness table, as Banks told Cooper during the interview. He said there is no reason for more delay:

"Immediately after learning of Dr. Ford's identity from news reports Sunday, committee staff started working to gather facts related to her claims. We've offered Dr. Ford the opportunity to share her story with the committee, as her attorney said yesterday she was willing to do. We offered her a public or a private hearing as well as staff-led interviews, whichever makes her most comfortable. The invitation for Monday still stands."

"Dr. Ford's testimony would reflect her personal knowledge and memory of events," Grassley said. "Nothing the FBI or any other investigator does would have any bearing on what Dr. Ford tells the committee, so there is no reason for any further delay."

A Justice Department spokesperson said the FBI had forwarded the initial letter from Sen. Dianne Feinstein on September 12 to the White House counsel's office in accordance with guidelines for conducting background checks.

The spokesperson noted that the allegation "does not involve any potential federal crime," and said if the FBI becomes aware of new information that raises suitability questions about a nominee or appointee, the agency will inform the White House, which is an existing agreement.

According to the spokesperson, the FBI "does not make any judgment about the credibility or significance of any allegation."

The letter comes after a day of uncertainty about whether the hearing scheduled for Monday would even take place, as Republicans continued to emphasize their repeated efforts to reach out to Ford.

Last week, news surrounding a private letter that had been sent to Feinstein, Democrat of California, raised potential questions about Kavanaugh's nomination. Feinstein later announced she had given the letter to the FBI. On Sunday, The Washington Post published a story that detailed Ford's personal account of an incident that Ford said took place when she and Kavanaugh were both in high school.

Ford alleges that while at a party, Kavanaugh pushed her into a bedroom along with his former classmate Mark Judge, and attempted to remove her clothes. She also alleges that Kavanaugh put her hand over her mouth when she attempted to scream. Kavanaugh has denied the incident, and he and Judge both deny being at the party in question.

"This is a completely false allegation. I have never done anything like what the accuser describes -- to her or to anyone. Because this never happened, I had no idea who was making this accusation until she identified herself yesterday," Kavanaugh said in a statement. "I am willing to talk to the Senate Judiciary Committee in any way the committee deems appropriate to refute this false allegation, from 36 years ago, and defend my integrity."

The public accusation led Grassley on Monday to announce that the committee would convene next Monday to give both Kavanaugh and Ford the opportunity to publicly testify, as both had indicated a willingness to do so.

But as of Tuesday afternoon, Senate Majority Leader Mitch McConnell said Republicans had yet to hear from Ford or her lawyer regarding their request for her to testify.

"What we're saying is there should be an investigation because that's the right thing to do," Banks said.

"She is prepared to cooperate with the committee and with any law enforcement investigation," she added.

Democrats have pushed back on the hearing. All 10 Democrats on the Senate Judiciary Committee sent a letter on Tuesday to FBI Director Christopher Wray and White House counsel Don McGahn arguing that the FBI should conduct an investigation prior to a hearing.

"The Committee should have the completed report before any hearing occurs and we ask that you take immediate steps to make sure that we have the FBI's report before we proceed," the senators wrote.

Feinstein said in a statement she supported Ford's proposal.

"We should honor Dr. Blasey Ford's wishes and delay this hearing. A proper investigation must be completed, witnesses interviewed, evidence reviewed and all sides spoken to. Only then should the chairman set a hearing date," Feinstein said in a news release.

"I hope that each and every one of us will immediately denounce the horrific treatment of Dr. Blasey Ford. That this brave woman is receiving death threats and has been forced to flee with her family is appalling and heartbreaking. This abuse must stop. We're better than this."

Senate Minority Leader Chuck Schumer, a New York Democrat, also supported Ford's proposal.

"Dr. Ford's call for the FBI to investigate also demonstrates her confidence that when all the facts are examined by an impartial investigation, her account will be further corroborated and confirmed," Schumer said in a statement. "Senate Republicans and the White House should drop their inexplicable opposition to an FBI investigation, allow all the facts to come out, and then proceed with a fair process in the Senate. Dr. Ford's life has already been badly disrupted by death threats and other intimidation. She deserves to be treated with respect and fairness by the Senate."

However, Sen. Orrin Hatch, a Utah Republican, countered Ford's proposal on Twitter Tuesday night, arguing, "The FBI does not do investigations like this."

"The responsibility falls to us," Hatch tweeted. "Chairman @ChuckGrassley has moved our committee vote to accommodate Dr. Ford's lawyer's offer on TV yesterday to have her client testify before the Judiciary Committee. We should proceed as planned."