The bureau will investigate aspects of the allegations made by Dr. Blasey, Ms. Swetnick and a third woman, Deborah Ramirez. She has said Judge Kavanaugh exposed himself to her during their freshman year at Yale. Ms. Ramirez’s lawyer, John Clune, said on Saturday that the F.B.I. had “reached out to interview Ms. Ramirez, and she has agreed to cooperate with their investigation.”

The F.B.I. plans to ask Mr. Judge about both Ms. Swetnick’s and Dr. Blasey’s allegations. Ms. Swetnick has said that Mr. Judge was at the parties where she saw Judge Kavanaugh mistreat women, and Dr. Blasey has said he was in the room when Judge Kavanaugh assaulted her. He has denied the allegations of both women.

The F.B.I. did not plan to question Ms. Swetnick herself, according to the two people familiar with the matter. Ms. Swetnick’s lawyer, Michael Avenatti, said on Twitter on Saturday that they had not heard from the bureau.

The four witnesses were Mr. Judge; Leland Keyser, a high school friend of Dr. Blasey’s who she said attended the party but was not told of the assault; P.J. Smyth, another party guest; and Ms. Ramirez, the Yale accuser.

The White House has asked that the F.B.I. share its findings after investigators complete those interviews, and at that point, Mr. Trump and his advisers would decide whether to have the accusations investigated further, the people said.

Led by Donald F. McGahn II, the White House counsel, Mr. Trump’s advisers are helping direct the scope of the background check, according to the senior administration official. Mr. McGahn shared the witness list with the F.B.I. but is working in concert with Senate Republicans, and senators considered key swing votes have had extensive input, the people said. Senators Susan Collins of Maine and Jeff Flake of Arizona have both said they want Mr. Judge questioned by the F.B.I.

The witness list did not extend to high school and college classmates who have said in interviews that Judge Kavanaugh drank heavily, including some who said he went beyond typical consumption.