Among these demands were a Thursday or later hearing date and a prohibition on the Senate Judiciary Committee enlisting female outside lawyers to assist them in taking Ford's testimony.
Sen. Susan Collins, a moderate Republican who earlier this week suggested Ford ought to testify next Monday, said in an interview Friday she was "appalled by the president's tweet".
"We'll do it on Wednesday; we expect the accuser before the accused and we do intend to have the counsel do the questioning", a Republican senator who asked not to be named told Politico.
Mr Grassley said he had consented to several other demands, including that Ms Ford be provided security and that Mr Kavanaugh not be in the hearing room when she testifies.
An FBI inquiry has been ruled out by both Trump and Republican lawmakers and the Senate Judiciary Committee has scheduled a hearing for both her and Kavanaugh on Monday.
During a speech to conservative activists at the Values Voters Summit in Washington on Friday, Senate Majority Leader Mitch McConnell appeared determined to stay the course.
"Women from every phase of Judge Kavanaugh's life, those who know him best, have stepped forward to say that the allegation being leveled against him is false and we are proud to stand with Brett", Fagen said.
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Asking Kavanaugh to go first is "not doable", according to the source, noting he would need to respond to any new accusation that comes up at the hearing. It looks like Grassley has the votes to move ahead with Kavanaugh's confirmation.
A group of 75 women also gathered on Friday to throw their support behind Kavanaugh in the midst of the scandal. Her lawyers say she's relocated her family due to death threats. On Monday, the Department of Justice weighed in on the matter, asserting Ford's allegation as it stands does "not involve any potential federal crime" warranting the FBI's involvement.
But onFriday, he changed tone, tweeting, "I have no doubt that, if the attack on Dr Ford was as bad as she says, charges would have been immediately filed with local Law Enforcement Authorities by either her or her loving parents".
According to Trump, the fact that Ford remained silent until now shows the incident probably never happened - even if this runs counter to what experts say is the typical reaction of sexual assault victims afraid or embarrassed to report.
The aggressive posture reflected Trump's fear that time is running out to get his hand-picked judge confirmed - thereby tilting the Supreme Court firmly to the right for years to come - before November elections when Republicans risk losing control of Congress. I don't think it's unusual at all that she didn't tell anyone back them.
His approval by the Judiciary Committee and the Republican-majority Senate appeared to be an nearly certainty until The Washington Post published its interview with Ford, who is now a California psychology professor. She said she was able to escape, but the Washington Post reported that she described the episode to a therapist in 2013 as a "rape attempt". The president has been accused of sexual misconduct by more than a dozen women - all liars, he has said. The proposal calls for Ford to testify first and Kavanaugh second.
But for Republicans, particularly in the Senate, it appeared that the only outcome potentially worse than confirming Kavanaugh is not confirming him. Organizers said it was signed by more than 1,000 former students.