President Trump said Wednesday he would have to “make a decision” about the future of Brett Kavanaugh’s nomination to the Supreme Court if accuser Christine Blasey Ford gives “credible testimony” about her allegation.

Speaking to reporters outside the White House, the president said if Ford appears before the committee “and makes a credible showing, that will be very interesting and we'll have to make a decision.”

“If she shows up, that would be wonderful,” Trump said. “If she doesn't show up, that would be unfortunate.”

Yet it’s not clear if the California professor who claims Kavanaugh sexually assaulted her more than 35 years ago will ultimately accept an invitation to testify on Capitol Hill, amid an impasse with Republicans, who have invited her to elaborate on her claims Monday.

In a new letter to Ford’s attorneys on Wednesday, Senate Judiciary Committee Chairman Chuck Grassley said he hopes Ford will still decide to appear.

“You have stated repeatedly that Dr. Ford wants to tell her story,” Grassley wrote to her lawyers. “I sincerely hope that Dr. Ford will accept my invitation to do so, either privately or publicly, on Monday.”

Ford says she doesn't want to appear until after an FBI investigation takes place, something law enforcement sources tell Fox News is not happening. Responding to Ford’s call for the FBI to investigate the allegations first, Grassley said it’s not the FBI’s “role to investigate a matter such as this.”

“The Constitution assigns the Senate, and only the Senate, with the task of advising the President on his nominee and consenting to the nomination if the circumstances merit,” Grassley said.

Meanwhile, a growing number of Republican senators are saying it’s time move ahead to a vote if Ford doesn’t show up for the planned hearing.

"If we don’t hear from both sides on Monday, let’s vote," Sen. Bob Corker, R-Tenn., tweeted.

South Carolina Sen. Lindsey Graham, a member of the Senate Judiciary Committee, dismissed the calls from Democrats for the FBI to investigate the allegations before Ford testifies. Graham said those calls are “not about finding the truth, but delaying the process till after the midterm elections.”

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“It is imperative the Judiciary committee move forward on the Kavanaugh nomination and a committee vote be taken ASAP,” he said.

Arizona Republican Sen. Jeff Flake, who had previously called for Ford to testify before voting on Kavanaugh's confirmation, indicated he's willing to move forward if she doesn't show.

"I think we'll have to move to the markup," Flake told CNN.

Trump said Wednesday he wants the Senate to move quickly, expressing sympathy for Kavanaugh and saying he has an “unblemished record.”

“This is a very tough thing for him and his family and we want to get it over with,” the president said.

Also on Wednesday, a spokesman for George W. Bush said the former president still stands by his support for Kavanaugh, who worked in his White House.

“Laura and I have known and respected Brett Kavanaugh for decades, and we stand by our comments the night Judge Kavanaugh was nominated: ‘He is a fine husband, father, and friend – and a man of the highest integrity. He will make a superb Justice of the Supreme Court of the United States,’” Bush said.

Late Tuesday, Kavanaugh’s accuser demanded a "full investigation" by the FBI before she attends any congressional hearing or "interrogation" into her accusations.

In response, 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.”

Republicans had repeatedly invited Ford and Kavanaugh to testify next week after delaying a planned Judiciary Committee vote that had been scheduled for Thursday. Kavanaugh accepted the committee's invitation, but Ford stayed quiet until Tuesday night.

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A federal law enforcement official told Fox News Ford's insistence on a FBI probe is "totally inappropriate."

"It's totally inappropriate for someone to demand we use law enforcement resources to investigate a 35-year-old allegation when she won't go under oath and can't remember key details including when or where it happened," the official said.

A highly-placed law enforcement source also said there won’t be a FBI investigation because there are no allegations a federal crime was committed. The bureau has also already conducted a background investigation.

“This is a political issue, not a law enforcement one,” the law enforcement source said.

The demand for a FBI investigation was also sure to add fuel to Republican claims that the allegations -- which were known to ranking Judiciary Committee Democrat Dianne Feinstein in July, but revealed to federal authorities just last Thursday -- are part of a concerted effort to stall Kavanaugh's nomination at the last minute.

"While Dr. Ford’s life was being turned upside down, you and your staff scheduled a public hearing for her to testify at the same table as Judge Kavanaugh in front of two dozen U.S. Senators on national television to relive this traumatic and harrowing incident," Ford's attorneys wrote to Grassley.

But Republicans on the Judiciary Committee directly disputed that claim Tuesday night, writing in a statement that Ford had been offered a chance to testify privately, and had never been told she would need to sit near Kavanaugh.

In the letter, Ford's lawyers went on to assert that Ford's family "was forced to relocate out of their home" and that "her email has been hacked, and she has been impersonated online."

The document concluded: "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 Judiciary Committee statement Tuesday night condemned any threats against Ford, but maintained that her request for an FBI probe was unfounded.

"The FBI has indicated to the committee and in public statements that it considers the matter closed," Grassley and other top Republicans wrote. "The FBI does not make credibility determinations. The FBI provides information on a confidential basis in order for decision makers to determine an individual’s suitability. The Senate has the information it needs to follow up with witnesses and gather and assess the relevant evidence."

The Republicans also disputed the suggestion in Ford's letter that the committee had not been accommodating: "Contrary to suggestions by Dr. Ford’s attorneys, the committee had no plans to place Dr. Ford and Judge Kavanaugh on a panel together, and never indicated plans to do so. Grassley’s staff offered Dr. Ford multiple dates as well as a choice of providing information in a public or private setting."

Meanwhile, Feinstein and Senate Minority Leader Chuck Schumer released statements backing Ford.

"We should honor Dr. Blasey Ford’s wishes and delay this hearing," Feinstein said. "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 had told Fox News earlier Tuesday that she could not be sure Ford was being entirely truthful.

"This is a woman who has been profoundly impacted by this," Feinstein said. "Now, I can't say everything's truthful. I don't know."

In a separate statement, Schumer said an FBI probe would be consistent with "precedent," adding, "Dr. Ford’s call for the FBI to investigate... demonstrates her confidence that when all the facts are examined by an impartial investigation, her account will be further corroborated and confirmed."

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Ford went public on Sunday, alleging that Kavanaugh forced himself onto her and covered her mouth in the 1980s, when Kavanaugh was 17 and she was 15. Ford did not mention the incident to others by her own admission until 2012, according to The Washington Post, when her therapist recorded her claim that four individuals had committed the assault.

Ford has since claimed that the therapist incorrectly transcribed that detail, and that she had said there were only two people in the room. Her husband has maintained that Ford mentioned Kavanaugh in the therapy sessions.

Ford also told The Post she could not remember in whose house the alleged incident occurred, the exact month of the episode, or why there was a gathering there.

Fox News' Gregg Re, Brooke Singman, John Roberts and Mike Emanuel contributed to this report.