“The Boehner-Netanyahu gambit has taken the relationship at the top to a new low,” Mr. Kurtzer said. “It has put Netanyahu into an anti-Obama camp, a Republican camp. He may think that he gains a political advantage by showing that Congress is on his side.”

Any final nuclear accord with Iran would require the United States to agree to repeal, over time, the sanctions that Congress has already imposed over the past decade for Iran’s nuclear program. As one senior administration official deeply involved in the talks said recently, “This whole set of discussions bolsters the hard-liners” in Tehran “who don’t believe we will reverse the sanctions no matter what Iran does.”

In the talks with Iran that have been underway in Vienna and Geneva, one of the key points of dispute has been Iran’s insistence that sanctions be lifted rapidly if an agreement is reached. Mr. Obama has resisted, saying he would use his own authority to suspend many of the sanctions in the early years of an accord. But those actions could be reversed by his successor. And ultimately, Iran wants a guarantee that the economic limits that have cut their oil exports by more than half and cut them off from the global financial system will be permanently ended.

The dispute with Israel has emphasized that an accord with Iran will require not one deal but three: one between Western and Iranian negotiators, one between Mr. Obama and Congress and one between Iran’s president, Hassan Rouhani, and the hard-line clerics and military officials who see the country’s nuclear program as the centerpiece of Iran’s ability to project power.

Mr. Netanyahu plays a major role in the second of those deals, as the shadow presence in the debate with Congress. But his effort may have also been directed at the Israeli electorate. “Netanyahu is using the Republican Congress for a photo-op for his election campaign and the Republicans are using Bibi for their campaign against Obama,” said Martin Indyk, a former ambassador to Israel, using the prime minister’s nickname. “Unfortunately, the U.S. relationship will take the hit. It would be far wiser for us to stay out of their politics and for them to stay out of ours.”