The two bills' sponsors — Sen. Mary Landrieu, D-La., and Rep. Bill Cassidy, R-La. — are heading toward a runoff election next month.
The incoming Congress will have the first black female Republican, a record number of women and minorities and the biggest GOP majority in the House since before World War II.
President Obama is planning to use his executive authority to offer temporary deportation relief to certain groups of immigrants. The Republicans say if he does, all bets are off.
While John Boehner has held the speaker's gavel, it often looked like members of his caucus were in control. With a gain of more than a dozen GOP seats, Boehner might have some more maneuvering room.