Sens. Ben Cardin and Bob Corker are hoping to move forward with the legislation.
The Senate Democratic leader said the law had turned out to be a "smashing success."
New amendments set up hurdles as lawmakers work on a path to passage.
Laura Burton Capps, the daughter of the retiring California Democrat, passes on a bid.
The senators said the VA has the authority to grant benefits to C-123 veterans.
Rep. Kurt Schrader (D-Ore.) has introduced legislation to cut salaries for members of Congress during future government shutdowns.
Senators are expected to debate Iran legislation Monday afternoon.
The Senate debates allowing Congress to review the Iran deal and the House takes up spending bills.
Del. Eleanor Holmes Norton (D-D.C.) thinks Iran should not be the only country to dismantle its nuclear program.
Rep. David Young (R-Iowa) missed a tax payment on his D.C. townhouse.