Washington (CNN) With the Supreme Court nomination of Brett Kavanaugh resting on a knife edge, President Donald Trump picked a Tuesday campaign rally to unleash his most direct broadside yet at his nominee's accuser.

His mocking of Christine Blasey Ford, a California professor who says Kavanaugh sexually assaulted her at a party in the early 1980s, drew cheers from the crowd. But it more important politically what reaction it will elicit from the handful of senators who remain undecided on Trump's nominee -- and whose support the President will need if Kavanaugh is to win confirmation to the nation's highest court.

"There's no time and no place for remarks like that," Republican Sen. Jeff Flake, a pivotal swing vote on the nomination, said on NBC's "Today" show Wednesday morning.

"To discuss something this sensitive at a political rally is just not right. I wish he hadn't had done it. It's kind of appalling," the Arizona Republican said.

Senate Majority Leader Mitch McConnell promised Tuesday that a vote would happen on Kavanaugh's nomination by the end of the week regardless of what the FBI finds in its investigation into the allegations against the nominee.

But McConnell's job of wrangling key Republican senators -- he can afford to lose only one vote from his caucus -- may have gotten harder after Trump decided the time was right to mock Ford's dramatic testimony to the Senate Judiciary Committee last week.

"I had one beer. Well, do you think it was -- nope, it was one beer," Trump said during the rally in Southaven, Mississippi. "How did you get home? I don't remember. How'd 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."

Tossed into this partisan brew Tuesday night was the news that the investigation has expanded to include interviews with Kavanaugh's friends about a July 1, 1982 party listed on the nominee's calendar. Democrats have focused on that date on Kavanaugh's calendar as a day when the alleged assault on Ford may have taken place, as she herself does not remember the date.

Senators have argued this week over the nomination and the supplemental investigation into allegations against Kavanaugh, which began last week after Flake called for the inquiry following a day of wrenching testimony by Ford, who claims that Kavanaugh sexually assaulted her when they were both in high school, and Kavanaugh himself, who vehemently denied the accusation.

The fate of the nomination is expected to hinge on a small group of undecided senators, including Flake as well as Republican Sens. Lisa Murkowski of Alaska and Susan Collins of Maine and Democratic Sen. Joe Manchin of West Virginia.

In an indication of the challenges Senate GOP leaders face in attempting to press ahead with the nomination, Flake suggested on Tuesday that the timing of a vote should be contingent upon what happens with the investigation, despite McConnell's promises for a vote this week.

Asked if he supports having a vote on Friday, Flake said, "Depends on where we are with the FBI investigation."

Conflict over scope of investigation, release of FBI report

Key points of contention among Democrats and Republicans have been the scope of the investigation and what will happen when the FBI report on the probe is released.

Following Flake's call for an investigation, the Senate Judiciary Committee last week requested a review of "current credible allegations" against Kavanaugh.

The panel's request immediately raised questions over which allegations would be deemed credible.

In addition to Ford's accusation, two other women have gone public with allegations involving the nominee.

Deborah Ramirez told The New Yorker that she remembers Kavanaugh exposing himself to her at a dormitory party, an allegation Kavanaugh has denied. And last week, Julie Swetnick, represented by attorney Michael Avenatti, presented the Senate Judiciary Committee with allegations of inappropriate behavior by Kavanaugh, accusing him of sexually aggressive behavior at alcohol-fueled parties when he was in high school. Kavanaugh issued an outraged denial in response to the allegation, calling it "ridiculous and from the Twilight Zone."

Democratic Sen. Amy Klobuchar of Minnesota told CNN's Jake Tapper over the weekend that the FBI should interview Swetnick, but prominent Senate Republicans have sought to undermine Swetnick's claims by suggesting she is not credible.

So far, the exact scope of the FBI investigation is not publicly known but the agency has interviewed Ramirez as well as three individuals who Ford has said were present at the gathering where the alleged incident involving Kavanaugh took place: Mark Judge, Patrick J. Smyth and Leland Keyser.

A dispute among senators has also emerged over how the release of the FBI report should be handled and whether its contents should be made public.

McConnell on Tuesday indicated that the report will not be made available to the public, saying that "only senators will be allowed to look at it." But Senate Democratic leader Chuck Schumer of New York pushed for a public release on Tuesday, saying "there is a growing consensus in the Senate that when the FBI investigation is complete, the findings should be released publicly with any personal information redacted."

Even some Republicans, including Senate Majority Whip John Cornyn of Texas and Sen. Lindsey Graham of South Carolina, have called for at least some of the FBI's assessment to be made public.

"There does need to be some sort of public statement -- if not the reports themselves -- because since the accusation had been made public it seems to me people are not going to be satisfied until (there's) some public statement about what the FBI supplemental background investigation shows," Cornyn said.

Whenever it does arrive on Capitol Hill, it appears that the report would remain initially behind closed doors. According to an aide to Senate Judiciary Chairman Chuck Grassley of Iowa, the committee had not been told yet when it will receive the supplemental background investigation as of Tuesday evening. When it arrives, the aide said, there will be one physical copy of the background investigation file that will be housed in a safe in committee offices. The only people who can read the FBI file are the 100 senators and nine staff members.

Democrats target Kavanaugh drinking; protests target key senators

As tensions flare, Democrats have suggested that Kavanaugh was not truthful during his testimony to the Senate Judiciary Committee, opening up a new line of attack against the embattled nominee.

During his testimony, Kavanaugh denied ever blacking out as a result of drinking. But over the weekend, a Yale classmate pushed back on that assertion , saying in a statement that "in denying the possibility that he ever blacked out from drinking, and in downplaying the degree and frequency of his drinking, Brett has not told the truth."

"Issues of credibility, of his telling the truth, constantly come up with Judge Kavanaugh," Schumer said on Tuesday, adding that "given what he said about his drinking, given what he said about his high school yearbook, all of which defy common sense and even defy objective truth, Americans have good reason to doubt this man's credibility."

In another sign of the tension surrounding the nomination and investigation, several Republican senators have been targeted by activists in protests captured on video.

Last week, Flake was confronted by protesters in a dramatic moment that aired live on CNN. The incident happened before he had called for an FBI investigation and shortly after he had announced he planned to support the nomination.

McConnell and Republican Sen. Bob Corker of Tennessee have also been confronted by activists in recent days.

"It was evident to me, this was all solely to get a video," Corker told reporters on Tuesday. "This is really not even about the Senate. To me, it's where the country is right now, and we as a nation have got to figure out a way to begin to unite ... right now there's just so many issues that separate us."