UNITED NATIONS — It has become the diplomatic big tease of the year, a rumored geopolitical rendezvous that, if not quite as momentous as Nixon and Mao in 1972, would still rank as a landmark encounter for two countries that have been estranged for more than three decades.

So, will President Obama actually shake hands with President Hassan Rouhani of Iran at the United Nations on Tuesday, when both men are scheduled to speak to the General Assembly?

On Monday, the White House insisted again that there was no meeting with Mr. Rouhani on the president’s schedule. But administration officials did nothing to dispel feverish speculation that the two leaders would find a way to bump into each other, whether in a hallway, in an elevator bank or by scraping their chairs together at lunch.

“We are open to engaging with Iran on a variety of levels,” Benjamin J. Rhodes, the deputy national security adviser, told reporters on Air Force One as Mr. Obama flew to New York. “This is not something we object to in principle. We will do so if we believe it is in our interest.”