It was the culmination of an odd week at the annual United Nations General Assembly, one that had none of the fireworks many had predicted. Mr. Trump, perhaps fearful that the Sept. 14 attack on Saudi oil facilities had placed the two countries on a path to conflict, perhaps distracted by the impeachment movement gathering steam in Washington, sounded strangely conciliatory toward Iran.

He did not call the Saudi attack an “act of war,’’ as his secretary of state, Mike Pompeo, had done. He did not try to organize a military response by allies. He kept saying the Iranians wanted “a deal,’’ without explaining what kind of deal he envisioned. In his speech to the General Assembly, Mr. Trump noted how the United States had befriended former enemies, a reference to Germany, Japan and Vietnam.

For his part, Mr. Rouhani, given several opportunities at both public and private events to comment on the prospect that Mr. Trump was in deep political trouble at home, decided to hold his tongue. Iran, he said with a smile, would deal with whatever president America sent to him. (One of Mr. Rouhani’s aides, however, sought out a reporter at a reception to ask whether Mr. Trump would survive the year in office.)

All week as Iranian officials and diplomats met their counterparts at the United Nations, the talk was centered on the details of how — not if — a meeting between Iran and the United States could be brokered. Yet both Mr. Trump and Mr. Rouhani seemed, independently, to conclude that meeting in New York was just too risky.

But both faced temptations. Mr. Trump, according to one Republican who spoke to him, saw a meeting as an opportunity to defuse pressure to retaliate for the attack on Saudi Arabia — and perhaps to commandeer headlines dominated by his possible impeachment.