Amendments filed by lawmakers last week include one that would require Iran to recognize Israel and another that would give any final nuclear deal the status of a treaty, which would require ratification by two-thirds of the Senate. Another proposal would require the release of American citizens detained in Iran as part of an agreement.

For Republican sponsors of the Iran measure, these amendments threaten to break the rare bipartisan spirit that pushed the bill unanimously out of the Foreign Relations Committee and even overcame White House objections. The bill’s unraveling would undermine the approach of Senator Mitch McConnell of Kentucky, the majority leader, and upset its many supporters.

“It’s important that this stay bipartisan,” said Senator Lindsey Graham, Republican of South Carolina. “We should not intermingle emotional amendments with this bill. I’m appealing to people, ‘Don’t throw this bill in a ditch.’ ”