But the White House would have to ask for it.
On Twitter, Grassley said committee "investigators are following up on the leads from Dr. Ford's allegations" and that "no other OUTSIDE investigation is necessary".
Mr Trump told reporters, "I can only say this: He is such an outstanding man".
Kavanaugh, who has categorically denied the claims, said he was willing to testify.
Chairman Chuck Grassley is also setting deadlines.
Ford's attorneys say she wants the FBI to investigate her allegation before she testifies.
Ms Blasey Ford claims Judge Kavanaugh sexually assaulted her when they were teenagers.
The confirmation fight comes just weeks before the November 6 congressional elections in which Democrats are seeking to take control of Congress from Trump's fellow Republicans, which would be a major blow to the president's agenda. "But the Federal Bureau of Investigation does not make a credibility assessment of any information it receives with respect to a nominee", Grassley argued. "She continues to believe that a full non-partisan investigation of this matter is needed and she is willing to cooperate with the Committee".
Grassley has rejected calls for an FBI investigation.
The White House under President George W Bush ordered that inquiry.
The development further roiled a confirmation process that once seemed smooth for Kavanaugh, whose confirmation to the lifetime post could consolidate the conservative grip on the top USA court.
"Fairness and respect for her situation dictate that she should have time to deal with this", Banks said. Clinton says she thinks Republicans are trying to put Ford "in the dock and try to rush this through".
Wednesday on CNN, she described the Senate's treatment of Ford as "callousness from my colleagues that I am totally appalled by". The White House. Judge Kavanaugh has not asked to have the FBI investigate these claims.
Anita Hill - whose allegation against Clarence Thomas has drawn endless parallels with the Ford-Kavanaugh controversy - says the accuser must not be hurried.
Others told their own stories of sexual assault.
Republicans control the Senate by only a narrow margin, meaning any defections within the party could sink the nomination and deal a major setback to Trump. "Very hard for me to imagine that anything happened". Now Republicans who pushed for Ford to get a chance to testify have hinted their patience has limits. "If she shows up that would be wonderful".
Senator Sheldon Whitehouse told the New York Times that Mrs Ford had been harassed and forced to leave her home for safety's sake.
He has called the allegation against Judge Kavanaugh "a drive-by shooting".
Those demands are "not about finding the truth, but delaying the process until after the midterm elections", Graham, a Judiciary panel member, said in a statement.
Ms Collins and Alaskan Senator Lisa Murkowski are moderate Republicans who have voted with Democrats in the past, making them possible swing votes when it comes to Judge Kavanaugh. Kavanaugh denies the accusation.
Susan Collins, who has been a crucial swing vote in the past, said it was not fair on Mr Kavanaugh if Mrs Ford declined to testify.
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