Will they or won't they? And what would it mean - if anything - for the future of the Middle East?
Rarely has a handshake been so eagerly anticipated. It is the talk of the town this morning. Even the most seasoned diplomats and hardened Middle East observers admit to a frisson of excitement at the prospect of an encounter between President Obama and his Iranian counterpart Hassan Rouhani.
However brief, it would be a moment of extraordinary symbolism, the first meeting between Iranian and American leaders since the fall of the Shah in 1979.
Amid the quagmire of Syria, the paralysis of the Palestinian-Israeli peace track, and the sickening violence playing out in Kenya, the world body is desperate for some good news this week.
Could a change of fortune in the Middle East come from a new reformist Iranian President prepared to surrender his nation's nuclear weapons ambitions?
But there are some analysts - and of course the Israeli government - who warn that the Iranian charm offensive is a cunning and brilliantly choreographed decoy. They believe that Iran will never give up its pursuit of nuclear weapons.
Rouhani, they warn, is being used by hardline forces and the Supreme Leader himself to persuade the West to lift sanctions. It is a mirage.
But when Rouhani addresses the General Assembly later today he will be speaking before an audience craving for an Iranian leader who shows flexibility and conciliation.
President Obama speaks before world leaders this morning several hours ahead of Rouhani. He can set the atmosphere - offering the hand of friendship but also insisting on concrete action from Teheran.
With thousands of journalists swarming over the United Nations, there is a single image we all want: The Handshake.
Perhaps it is naive - and no doubt the negotiations with Iran will be tortuous and technical - but surely the White House won't disappoint a war-weary world desperate for some good news.
If it happens - and assuming an official photographer is on hand - the encounter will be one of the defining images of 2013.
For President Obama it would also be an extraordinary shift from a few weeks ago, when the US was on the brink of attacking Syria. Now suddenly there is the rarest of moods at the United Nations: Optimism.