Reporting by CBS News' Alan He, Kathryn Watson and Major Garrett

Senate Majority Leader Mitch McConnell filed a motion Wednesday for a cloture vote on Supreme Court nominee Brett Kavanaugh, setting in motion the procedure that would usher in the first vote sometime Friday and the vote on the full Senate floor on Saturday. McConnell also announced that the FBI's updated report on Kavanaugh was ready.

On the Senate floor, McConnell said, "This evening, the Senate will receive the results of the FBI's supplemental background investigation of Judge Brett Kavanaugh," adding that senators would "have the opportunity to review the investigators' records."

Early Thursday, Senate Judiciary Committee Chairman Chuck Grassley, R-Iowa, tweeted that the panel had received the report after White House Deputy Press Secretary Raj Shah said in a statement that the White House had transmitted the report to the Senate.

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The White House contends that Senators have "ample time" to review the seventh background inquiry into Kavanaugh's record with McConnell's cloture filing. Those close to the situation now anticipate a confirmation vote to come some time Saturday. Precise scheduling will come from Leader McConnell.

"This is the last addition to the most comprehensive review of a Supreme Court nominee in history, which includes extensive hearings, multiple committee interviews, over 1,200 questions for the record and over a half million pages of documents. With this additional information, the White House is fully confident the Senate will vote to confirm Judge Kavanaugh to the Supreme Court," Shah added.

Describing the review process as standard procedure, McConnell said designated Judiciary Committee staffers with the required clearances would be authorized to brief members. That was expected to begin Thursday morning.

"There will be plenty of time for Members to review and be briefed on this supplemental material before a Friday cloture vote," McConnell said.

President Trump tweeted after McConnell's motion that there is "such enthusiasm and energy" for Kavanaugh. "Look at the Energy, look at the Polls. Something very big is happening. He is a fine man and great intellect. The country is with him all the way!" Mr. Trump wrote.

Wow, such enthusiasm and energy for Judge Brett Kavanaugh. Look at the Energy, look at the Polls. Something very big is happening. He is a fine man and great intellect. The country is with him all the way! — Donald J. Trump (@realDonaldTrump) October 4, 2018

Attorneys for Christine Blasey Ford, the Kavanaugh accuser who testified before the Senate last Thursday, said that the FBI's investigation "did not include an interview of Dr. Christine Blasey Ford," and added that they were "profoundly disappointed that after the tremendous sacrifice she made in coming forward, those directing the FBI investigation were not interested in seeking the truth."

Sen. Ben Sasse, R-Nebraska, became emotional on the Senate floor before McConnell filed for cloture. Mentioning his two daughters, he said he wanted them to be taken seriously if they came forward with accusations of sexual assault.

"We all know that the president cannot lead us through this time," Sasse said. "We know that he's dispositionally unable to restrain his impulse to divide us. His mockery of Dr. Ford in Mississippi was wrong, but it doesn't really surprise anyone. It's who he is. Similarly, it was wrong last week when he said -- quote -- that if the attack on Dr. Ford was as bad as she says, charges would have been immediately filed with local law enforcement authorities."

Earlier Wednesday, Senate Majority Whip John Cornyn, R-Texas, told reporters that he hoped that some material would be released to the public, in order to clear Kavanaugh's name.

"And I hope some conclusion or some summary will be made public because I hate to see Judge Kavanaugh be subjected to these allegations in public, which have to be embarrassing and destructive and not see the vindication if in fact that's what we see made public," Cornyn added.

Whether that report will make its way into the public is unclear.

A handful of Republican senators — including the staunchest defenders of Mr. Trump and Kavanaugh — criticized the president for the testimony of Kavanaugh accuser Christine Blasey Ford at a rally in Mississippi Tuesday night.

"How did you get home? 'I don't remember,'" Mr. Trump said at the rally in Southaven, appearing to alternate between acting as a questioner and giving an impression of Ford. "How did you get there? 'I don't remember.' Where is the place? 'I don't remember.' How many years ago was it? 'I don't know.'"

Sen. Lindsey Graham, R-South Carolina, said he'd tell the president to "knock it off."

Sen. Lisa Murkowski, R-Alaska, a key swing vote, called the president's remarks "wholly inappropriate."