The administration, which originally resisted any role for Congress, sounded notes of acquiescence Thursday. “We have been clear that the bill passed out of the Senate Foreign Relations Committee last month was the kind of reasonable and acceptable compromise that the president would be willing to sign,” said Bernadette Meehan, a spokeswoman for the National Security Council, adding, “We were pleased to see that what just overwhelmingly passed the Senate stayed true to that bipartisan compromise and we are hopeful that the House will similarly protect this compromise bill.”

Ultimately, it was less White House agitation than Republican infighting that prevented significant amendments to the bill, leaving some members deeply unhappy that they were unable to weigh in further on a matter that many said was the most significant of their careers.

Image Senator David Vitter, Republican of Louisiana, center, on Capitol Hill in April. Credit... Doug Mills/The New York Times

Republicans and Democrats had gingerly worked out a deal to allow votes on a few amendments. But some Republicans, angered that their amendments were going to be curtailed or rejected, employed procedural maneuvers that forced Mr. McConnell to take the bill to the floor without any significant amendments.

“I am deeply disappointed by the direction this debate has taken,” said Senator Marco Rubio, Republican of Florida and a candidate for president, who had wanted to debate an amendment that would require Iran to recognize Israel.

But in the end, a bipartisan accord that seemed nearly impossible in the cantankerous Congress just a few months ago came together by a formidable margin.

“Let me be clear,” Senator Mitch McConnell, Republican of Kentucky and the majority leader, said on the Senate floor as he encouraged members to approve the bill while noting the procedural fights that hobbled the process. “Our response to this should not be to give the American people no say at all,” adding, “make no mistake that this will not be the end of the story.”