WASHINGTON—Senior Republicans said Monday they wouldn’t restrict who the FBI could question as part of its investigation into sexual-assault allegations against Supreme Court nominee Brett Kavanaugh, but said they planned to begin floor votes this week on his confirmation.

President Trump said Monday that he wanted a “comprehensive” inquiry by the Federal Bureau of Investigation, although he also made clear he desired quick action by the agency. Senate Majority Leader Mitch McConnell (R., Ky.) said the Senate would begin...