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.
Both Corker and Cardin made their opening salvos in support of the legislation.
House GOP leaders denied a vote on the proposal last year.
A Cruz spokeswoman suggested a procedural vote was "the only vote that mattered."
The vote came more than 160 days after she was nominated by President Obama.