WASHINGTON—Senate Majority Leader Mitch McConnell pressed Tuesday to hold a vote as soon as possible on Judge Brett Kavanaugh, with the FBI expected to wrap up its probe into allegations of sexual assault and misconduct by the Supreme Court nominee as soon as Wednesday.

Mr. McConnell was vague on the timing of his next steps. While he made clear the FBI’s findings wouldn’t derail plans for a vote, he retained flexibility to address the concerns of three undecided GOP senators—Jeff Flake of Arizona, Susan Collins of Maine and Lisa Murkowski of Alaska.

Given the GOP’s thin Senate majority, Mr. McConnell would need support of two of those three lawmakers if all Democrats vote against the judge.

“What I can tell you with certainty—we’ll have an FBI report this week, and we’ll have a vote this week,” Mr. McConnell, a Kentucky Republican, told reporters.

At a political rally in Mississippi late Tuesday, President Trump mocked the testimony of Christine Blasey Ford, a research psychologist who alleged Judge Kavanaugh sexually assaulted her at a Maryland house party when they were teenagers. Judge Kavanaugh, now 53 years old, has categorically denied the allegation.