The Senate late Friday backed withholding the White House budget director's pay if the president is late delivering a budget.
The amendment would also withhold the pay of the deputy director and deputy director for management of the Office of Management and Budget.
President Obama was supposed to deliver his budget by the first Monday of February under the law, but his 2014 budget is now slated to come out in April.
Cornyn told the Senate that "the president has rendered himself entirely irrelevant," in the budget process since the House and Senate will have acted first.
He noted that after Congress enacted a "No Budget, No Pay" law this year, the Senate brought a budget to the floor for the first time in four years.
No Democrats rose to defend Obama.
Obama has nominated Sylvia BurwellSylvia Mathews BurwellOvernight Healthcare: GOP chairman to introduce pre-existing condition bill ObamaCare enrollment hits 11.5M for 2017 Obama, Dems eyeing strategy to defend ObamaCare MORE to take over has his budget director, a position that was vacated in 2012 when Jack LewJack LewOne year later, the Iran nuclear deal is a success by any measure Chinese President Xi says a trade war hurts the US and China Overnight Finance: Price puts stock trading law in spotlight | Lingering questions on Trump biz plan | Sanders, Education pick tangle over college costs MORE became Obama's chief of staff. Lew is now the Treasury Secretary.
Acting Director Jeff Zients is expected to present Obama's budget when it comes out April 8.