Separately, the Senate Judiciary Committee made public late on Sunday a previously unreleased interview with Kavanaugh from September 26, before a public hearing with Ford, in which he denied all the allegations against him and committee Democrats declined to ask questions, saying they felt the FBI should investigate the allegations.
Flake, a retiring Arizona senator who sits on the Senate Judiciary Committee, on Friday broke party lines and said he would not cast a final vote to confirm Kavanaugh until the FBI investigated the groping allegations against him.
Speaking to the issue of the scope of the FBI's investigation, White House press secretary Sarah Huckabee Sanders said White House counsel Don McGahn, who is managing Mr Kavanaugh's nomination, "has allowed the senate to dictate what these terms look like, and what the scope of the investigation is". We're not micromanaging this process.
Democrats have been alarmed at reports in U.S. media the White House was limiting who could be interviewed under the investigation.
"It's a Senate process. It has been from the beginning, and we're letting the Senate continue to dictate what the terms look like", Sanders said. "The FBI is not tasked to do that".
The classmate's attempts to talk with the Federal Bureau of Investigation about this, however, have been one dead end after another.
"I think that will be the scope of it".
Senator Amy Klobuchar, a Democratic member of the Judiciary Committee, told CNN that she was "very concerned" by the reported White House constraints on the inquiry.
Sen. Susan Collins said Sunday she is confident in the investigation and "that the FBI will follow up on any leads that result from the interviews".
Republicans, who are trying to retain control of the US Congress in November elections, are seeking to balance their desire for another conservative justice on the court with sensitivity about how they handle sexual misconduct allegations amid the reverberations of the #MeToo movement.
In a separate report, NBC quoted a senior United States official and a second source as saying the limits imposed by the White House on the inquiry remained in place, despite Trump's tweet.
Senate Republicans compiled the list of four witnesses and Senate Republican leader Mitch McConnell shared it with the White House, the official and another source familiar with the matter told Reuters.
Carl Tobias, a law professor at the University of Richmond, said it would not be unlawful for the White House to restrict the investigation's scope because the FBI is under the executive branch.
"I thought it was going to be an investigation, but instead it seems it's just an alibi for Republicans to vote for Kavanaugh", he said. Judge has said he "categorically" denies the allegations.
Another committee member, senator Lindsey Graham, said testimony would be taken from Ms Ramirez and Mr Kavanaugh's high school friend Mark Judge, who has been named by two of three women accusing Mr Kavanaugh of sexual misconduct.
Rachel Mitchell was appointed by Senate Republicans to question Ford during her testimony before the Senate Judiciary Committee in order to avoid the optics of 11 men questioning a woman about an alleged sexual assault.
He revisited the "scope" question later Saturday on Twitter, writing in part, "I want them to interview whoever they deem appropriate, at their discretion".