WASHINGTON—Waiting his turn to testify last week before the Senate Judiciary Committee, Judge Brett Kavanaugh had a private word with White House counsel Don McGahn, who has been shepherding his Supreme Court nomination. Mr. McGahn cleared the holding room of staff, people familiar with the matter said.

Christine Blasey Ford had just told senators that he sexually assaulted her when they were teens, and White House aides worried her account was compelling. Sitting with Judge Kavanaugh and his wife, Mr. McGahn told him he shouldn’t...