The White House said on Thursday a meeting was possible, the first between US and Iranian presidents since the 1979 Islamic Revolution.
"It's possible, but it has always been possible," White House spokesman Jay Carney said. "The extended hand has been there from the moment the president was sworn in."
It looks more likely to be a handshake and brief exchange of pleasantries - probably in the UN building - rather than a formal meeting where the leaders could talk at greater length
With conciliatory overtures and gestures emanating from Iran's ruling echelon at a surprising pace in recent days, the White House is looking for the right balance in forming a response.
Obama eventually wants to encourage Iran to make concessions in talks over its nuclear program. But if he embraces Tehran too warmly before it takes concrete actions, he would risk criticism that he is fumbling another foreign policy issue after struggling to handle crises over Syria and Egypt.
SIGNS OF WARMING
Iran's rhetoric has softened markedly since Hassan Rowhani took office in August. Recent gestures include a promise never to develop nuclear weapons, tweeted greetings on the Jewish New Year and the release of prominent political prisoner and rights lawyer Nasrin Sotoudeh.
On Thursday, Hassan Rowhani published an opinion piece in the Washington Post urging other leaders "to respond genuinely to my government's efforts to engage in constructive dialogue."
For its part, the White House said this week Obama had written Hassan Rowhani to convey the message "that the US is ready to resolve the nuclear issue in a way that allows Iran to demonstrate that its nuclear program is for exclusively peaceful purposes."
A US official said the White House hoped to engineer a handshake in the UN building between the two leaders, but by no means a full meeting, and a second official also bet on a handshake, while saying there were currently no such plans.
Regardless of whether Obama and Hassan Rowhani shake hands, the more serious issue is whether both countries are ready to get into a direct bilateral discussion.
The United States suspects Iran of using its civilian nuclear program as a cover to develop atomic weapons, something it sees as a threat to Israel and to oil-producing US allies in the Gulf. Iran denies that, saying its nuclear program is for purely peaceful purposes.
A decade of negotiations between Iran and the West has yet to resolve the dispute and the United States has said it would not take any option off the table - code for a possible military strike - in dealing with Iran's nuclear program.
Speeches by Obama and Hassan Rowhani, who address the United Nations next Tuesday, will attract scrutiny for signs of a thaw. Another closely watched address will be that of Prime Minister Benjamin Netanyahu, who views a potential Iranian bomb as an existential threat to Israel and is wary of Iran's new tone.
Hassan Rowhani may extend what many analysts regard as a charm offensive by distancing himself from remarks by his predecessor, Mahmoud Ahmadinejad, who was widely vilified in the West for doubting the Holocaust and questioning Israel's right to exist.
Obama's speech must strike a balance, analysts said, between showing a readiness to engage Iran - a message he conveyed in his first week as president in 2009 by saying he would extend a hand if they would "unclench their fist" - and stressing that talks could not be endless and Iran must curb to its nuclear program.
In so doing, Obama needs to keep the door open to talks while protecting himself from attacks from conservatives who may regard his willingness to talk as weakness, particularly after his recent decision not to bomb Syria.
Elliott Abrams, who served under former Republican President George W. Bush and is now at the Council on Foreign Relations think tank, said Obama was right to test whether Iran was willing to negotiate but should avoid an encounter with Hassan Rowhani himself.
Saying the two are not equals because Hassan Rowhani serves under Iranian Supreme Leader Ali Khamenei, Abrams said: "Such a meeting is likely to be read in Tehran as showing how anxious Obama is for a deal. It ought to be avoided."
Abrams also said Obama had undercut his leverage with Iran by striking a diplomatic deal with Russia to try to eliminate Syrian nuclear weapons rather than launching a military strike that he appeared poised to order in late August.
"What happened with regards to Syria (suggests) that the Americans don't want any kind of military engagement, so all options are not on the table with regards to Iran," he said, saying that might make Israel more likely to strike Iran if the Jewish state thinks Washington is not engaged.
While there has been speculation of talks between the two presidents or between US Secretary of State John Kerry and Iranian Foreign Minister Jawad Zarif during the UN meetings, current and former US officials said lower-level contact might make more sense.
"That's often the way they start because you're not really sure what you're dealing with," said a third US official. "You can survive a lower-level meeting that doesn't work, but you can't survive a higher-level that doesn't work."
The United States has several potential candidates to lead the talks, including Wendy Sherman, undersecretary of state for political affairs, who currently leads US nuclear negotiations with Iran, and Bill Burns, deputy secretary of state, who is a past negotiator with Iran and a Middle East expert.
"The level and the negotiator will not be difficult to arrange ... the formal trappings, they'll figure out," said Dennis Ross, a former senior White House official under Obama now at the Washington Institute for Near East Policy think tank.
"If the decision has been made to do it, you cut through that. If they are still fencing around trying to determine how to do it, then that is an indication that there isn't quite the readiness there otherwise would appear to negotiate," he added.
Karim Sadjadpour, an Iran expert at the Carnegie Endowment for International Peace, said a genuine rapprochement between Iran and the United States was unlikely but that an Barack Obama-Hassan Rowhani handshake "could open a path toward detente."
"As long as Ayatollah Khamenei remains supreme leader of Iran, this is the best group of interlocutors that the US will ever have to work with in Tehran, particularly Foreign Minister Zarif," Sadjadpour said.
He argued that Hassan Rowhani and his foreign minister might succeed in impressing other Western nations with their more conciliatory tone and that could, over time, make it harder for the United States to sustain economic sanctions on Iran.
"I think the double-edged sword Hassan Rowhani and Jawad Zarif present to the United States and Israel is that Iran is now easier to engage, but more difficult to isolate," he added.