WASHINGTON—A federal mortgage rescue to help hundreds of thousands of homeowners avoid foreclosure is in limbo, with the Senate about to pass a bill that has no chance of winning House approval.

Democratic divisions over small but significant details are delaying the plan, despite keen interest by lawmakers in both parties to enact election-year help targeted to the housing crisis at the root of voters’ anxiety. Complicating the picture is a White House veto threat.

The measure is expected to pass easily as early as Thursday in the Senate, where it consistently has draw...