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.
Sen. Harry Reid said Monday's vote on going to conference is "an exercise in futility."
The Utah Republican said lawmakers should coincide the "threat of sanctions" with Israeli Prime Minister Benjamin Netanyahu's speech