The Energy Department would have 30 days to approve applications under the bill.
The House on Tuesday passed legislation to enhance law enforcement efforts against human trafficking.
GOP freshmen haven't had a chance to vote to repeal the law yet.
Rep. Frank Guinta (R-N.H.) said his proposal would prevent inflated bureaucracy.
The new legal action will focus on Obama's executive actions on immigration.
Senate Majority Leader Mitch McConnell (R-Ky.) is a co-sponsor.
Six more bills are on tap for consideration on Tuesday.
A border security bill has also been dropped from the schedule due to the shortened workweek.
Rep. Ted Poe (R-Texas) said President Obama's threat during his State of the Union address to veto any legislation imposing more sanctions on Iran was counterproductive.
The lawmakers have criticized the Republican Study Committee as too cozy with leadership.