And in what appeared to be an extraordinary diplomatic gesture captured on video at a meeting of the United Nations Security Council on Thursday, the American ambassador, Kelly Craft, walked over to chat with her Iranian counterpart, Majid Takht-Ravanchi. While their aides sought to play down its significance, such a public encounter has rarely been seen in recent years between American and Iranian diplomats at the United Nations.

Kazuo Takahashi, a professor emeritus at the Open University of Japan who specializes in Middle East politics, said Mr. Rouhani hoped to show with his visit that Iran was still respected and not isolated, “and that he may even be able to get some concessions from Japan or the United States.”

Mr. Takahashi speculated that Mr. Rouhani had passed a message to Mr. Trump that “he might be willing to talk,” citing the recent prisoner exchange as evidence.

Iran has publicly distanced itself from any suggestion of a quick diplomatic breakthrough with the United States. Mr. Rouhani has said that the purpose of his visit was to discuss the security of the Middle East and to calm tensions in the Persian Gulf. On Monday, Ali Rabiei, a government spokesman, said Mr. Rouhani’s trip was “not related to negotiating with the U.S., but we welcome Japan’s messages and new efforts.”

Still, the meeting with Mr. Abe had been requested on relatively short notice by Iran, according to a Japanese government official briefed on the issue. Mr. Takahashi said that was an indication the Iranians had “an urgent, important message they want to give to Mr. Trump.”

In Iran, supporters and opponents of Mr. Rouhani were scrutinizing the trip for signs of an opening with the Americans. Many Iranians want a diplomatic solution to the sanctions, as does Mr. Rouhani’s political party.

But Iran’s supreme leader, Ayatollah Ali Khamenei, has publicly ruled out talks with the United States as long as sanctions remain in place.