WASHINGTON—The immigration debate that begins Monday in the Senate promises to be the rarest of things in Washington: a freewheeling, open-ended battle over one of the thorniest subjects, with the expected outcome completely unclear.

Majority Leader Mitch McConnell (R., Ky.) will begin debate on a shell bill that has no immigration provisions, to be filled in with senators’ amendments. The process could take days or weeks, depending on the number of amendments offered and the amount of time Mr. McConnell wishes to devote to...