Speaking at a news conference at the White House on the reworked free trade agreement between the US, Canada and Mexico, Mr Trump said he wanted the investigation to "go quickly".
The most senior Democrat on the Senate's Judiciary Committee, Dianne Feinstein, has sent a letter urging the White House to release the written directive sent by President Trump to the FBI launching the investigation.
Kavanaugh has denied Ramirez's account that he exposed himself to her and thrust his genitalia in her face during a party at a Yale dormitory when they were both students there, as reported September 23 by The New Yorker.
Trump said Kavanaugh has been treated horribly. Kavanaugh's testimony contained some iffy statements, like his definitions for "Devil's Triangle' and "boofing", Kimmel said".
"On many occasions I heard Brett slur his words and saw him staggering from alcohol consumption", Ludington wrote.
Sen. Susan Collins of ME - one of three Republican senators who was instrumental last week in holding up Kavanaugh's confirmation vote - said she had extensive conversation with the White House counsel's office and is "confident that the FBI is doing a thorough investigation and that it will be helpful to us as we make our decisions".
Trump on Friday had ordered the new FBI investigation into sexual assault allegations against Kavanaugh, as the Senate delayed a final vote on his confirmation as the key ninth member of the Supreme Court.
"I think we're all pretty confident, given that we've been through this process a number of times, but we would assess that at that point", Ms Sanders said.
Because this is an inquiry and not a criminal investigation, the FBI will not compel anyone to talk to them.
White House counsellor Kellyanne Conway said the investigation will be "limited in scope" and "will not be a fishing expedition".
Mr Ludington added that while youthful drinking should not condemn someone for life, he had nevertheless been concerned about Mr Kavanaugh's statements under oath before the Senate Judiciary Committee.
"The Supreme Court is one of the last bastions of trust".
Mr Kavanaugh has denied the allegations.
"Our Democratic colleagues have accused Kavanaugh of being a rich, drunk, lying sexual predator and the six FBI investigations he's been through don't support that". After hearing from Ford and Kavanaugh last week, and meeting with sexual assault survivors, Collins encouraged calls for an FBI investigation before the Senate votes.
GOP leaders reluctantly agreed to a limited investigation lasting no longer than one week due to their razor-thin majority in the Senate.
And even as the FBI explored the past allegations that have surfaced against Kavanaugh, another Yale University classmate came forward to accuse the federal appellate judge of being untruthful in his testimony to the Senate judiciary committee about the extent of his drinking in college.
"It is truth that is at stake, and I believe that the ability to speak the truth, even when it does not reflect well upon oneself, is a paramount quality we seek in our nation's most powerful judges". Mr Trump's fellow Republicans control the Senate by a narrow 51-49 margin.
The FBI is expected to spend the coming days investigating the sexual misconduct claims against Kavanaugh. "Instead, it seems that the Republican goal is to be able to say there was an investigation and it didn't change their view, while the Democrats hope for incriminating evidence to derail the nominee".