The Supreme Court is hearing oral arguements on a challenge to the health law.
Senators are expected to take a procedural vote Wednesday on overturning the veto.
Sen. Mitch McConnell (R-Ky.) said lawmakers could take up the legislation next week.
The Senate will vote to roll back a rule making it easier to hold union elections.
The legislation would let Congress weigh in on any Iran nuke deal.
Sen. Lindsey Graham (R-S.C.) suggested the Israeli official would be able to help explain what a good or bad deal on Iran's nuclear program would look like.
The Kentucky Republican said the administration is putting politics above policy.
Sen. Chuck Schumer (D-N.Y.) said Boehner could solve the legislative stalemate "with the snap of his fingers."
Democrats easily blocked the measure in a 47-43 procedural vote.
Sen. John Barrasso (R-Wyo.) said Obama has "mishandled" the negotiations.