(CNN) -- Key Republican senators have called the White House over the past 24 hours to make clear that they expect a fulsome investigation as the FBI reviews allegations against Supreme Court nominee Brett Kavanaugh, CNN has learned.

Following questions about the scope and extent of the FBI investigation, a White House official also told CNN that the White House has made it clear to the FBI that agents are not limited in their expanded background search.

CNN was told that on Friday, Senate Majority Leader Mitch McConnell informed the White House that Republican Sens. Jeff Flake of Arizona, Lisa Murkowski of Alaska and Susan Collins of Maine wanted the FBI to interview four witnesses: Mark Judge, Leland Keyser, Patrick J. Smyth and Deborah Ramirez, the first three possible witnesses to an allegation of sexual assault by Christine Blasey Ford while she and Kavanaugh were in high school. Ramirez has separately accused Kavanaugh in a report from The New Yorker of inappropriate sexual behavior related to his time at Yale. Kavanaugh has denied all allegations against him.

But the senators whose votes hold the key to confirmation wanted the White House to know they expect those interviews to be a start, not necessarily the full extent of the investigation.

Ford, Kavanaugh's first public accuser, has said that Judge, Keyser and Smyth were present at the party where she says Kavanaugh assaulted her in the early 1980s.

The FBI has initiated its investigation into allegations against Kavanaugh at the direction of the White House after Flake last week -- with the backing of Murkowski and Collins -- called for a delay in consideration of the nomination so that the FBI could undertake a review.

The Senate Judiciary Committee has said that the investigation would be "limited to current credible allegations" -- raising questions over exactly what would be investigated given that Kavanaugh faces multiple allegations.

On Monday, Collins said she is "confident that the FBI will follow up on any leads that result from the interviews" when asked if the senator is satisfied with the scope of the inquiry.

Appearing at the Forbes Under 30 Summit in Boston, Flake said on Monday that he wants the FBI to conduct a "real investigation" and not one that "just gives us more cover."

"We certainly want the FBI to do a real investigation and we are working to make sure that that happens," he said. "I've had discussions, many yesterday with my colleagues, with the White House counsel's office. My staff is following up as well. I had one of those conversations just five minutes ago to make sure that ... any current credible allegation that has been made is fully investigated."

Flake added, "It does no good to have an investigation that just gives us more cover, for example," and said that, "we actually need to find out what we can find out."

On Monday, President Donald Trump decried what he described as "trauma" inflicted upon Kavanaugh, but nevertheless said he wants to see the FBI "do a very comprehensive investigation."

With the investigation underway, Senate Republican leaders appear intent on moving forward with the nomination as swiftly as possible.

McConnell, accusing Democrats of moving the goalposts on the Kavanaugh nomination, said on Monday that the time for "delay and obstruction" has come to a close.

On the Senate floor, he said, "we will be voting this week."