WASHINGTON—Senate Majority Leader Mitch McConnell set up a critical procedural vote Friday on Supreme Court nominee Brett Kavanaugh as senators prepared to scrutinize an FBI report into allegations of sexual assault against the nominee.

Mr. McConnell (R., Ky.) said the Senate would receive the FBI supplemental background investigation into Judge Kavanaugh late Wednesday. He said senators would have time starting Thursday morning to review bureau’s findings before voting later this week.

Friday’s...