WASHINGTON—The threat of a partial government shutdown intensified as senators signaled opposition to a short-term spending bill that the House passed Thursday evening.

Senate Democrats indicated on Thursday they had the votes to block the measure to keep the government funded through Feb. 16, according to multiple congressional aides. At least two Republican senators also opposed the bill. Republicans control 51 Senate seats but need at least 60 votes to clear procedural hurdles, a tall order if Democrats stay unified and...