Ebrahim Noroozi / AP Iranian President-elect Hasan Rouhani visits Ayatollah Khomeini's shrine, just outside Tehran, on June 16, 2013

Updated July 23, 2013, 10:50 a.m.

Last Saturday, incoming Iranian President Hassan Rouhani tweeted approvingly that “131 [U.S.] Congressmen have signed a letter calling on President #Obama to give peace a chance with Iran’s new president #Rouhani.” He followed it later in the day with a tweet announcing, “National Security & Foreign Policy Committee of Iran’s Majlis [parliament] to look into potential change in US approach to Iran presstv.ir/detail/2013/07…”

Set aside for a moment the fact that the incoming Iranian President tweets. Rouhani’s move to highlight a largely symbolic letter by a powerless group of House Democrats is part of his broader effort to shore up Iranian political support for improved relations with the U.S. And it shows just how big an atmospheric shift is under way in Iranian diplomacy in the wake of Rouhani’s election. Whether the change is just atmospheric, or is something more substantive, is a key question for President Obama as he decides how to respond to Rouhani’s outreach.

As things stand, the U.S. and Iran are potentially on a course for war in Obama’s second term. Last year, Obama promised military action, if necessary, to prevent Iran from getting a nuclear weapon. Iran says it has only peaceful intentions, but has accelerated its nuclear program over the past year even as it has been careful not to go far enough to trigger an Israeli attack.

(PHOTOS: Iranians Cast Vote in an Uneasy Election)

Since he won the election with a surprising 50.7% in the first round of voting on June 14, Rouhani has taken a much softer tone on the U.S. than his predecessor, Mahmoud Ahmadinejad. On June 18, he tweeted a “flash from the past” picture of himself visiting a U.S. field hospital set up to treat victims of a 2003 earthquake in Iran. At his first press conference after the election, Rouhani said the U.S. and Iran need to heal the “very old wound” between them and “find solutions to past issues.” Says the Majlis’ National Security and Foreign Policy Committee deputy chairman Ahmad-Reza Dastgheib: “It is clear that Mr. Rouhani is seeking to change Iran’s diplomatic process, and that he will make efforts in this regard.”

There are two opposing interpretations of Rouhani’s outreach in Washington. Doves approvingly cite Rouhani’s suspension of the Iranian nuclear program in 2003 when he was Tehran’s top negotiator and say now that he’s President he may be willing to do so again, a key U.S. demand. They say the U.S. needs to make a big gesture to show it is open to diplomatic progress and to help bolster him against Iranian hard-liners. Former national-security advisers Brent Scowcroft and Zbigniew Brzezinski have endorsed a call for Obama to meet directly with Rouhani this fall when the two men are in New York City for the U.N. General Assembly. “We have to take advantage of this moment,” says Joe Cirincione, president of the Ploughshares Fund. Cirincione says of an Obama-Rouhani meeting “a gesture like that to make it easier for Rouhani to make the deal.”

Hawks suspect Rouhani is just a clever diplomatic tactician and view the calls for softness as a dangerous misstep. They suspect Rouhani’s outreach is a ruse: an attempt to get relief from crippling sanctions by changing the tone of Iranian diplomacy without making any major concessions on the nuclear front. “They’ve built up a capability that is very impressive,” says Joseph DeTrani, former ambassador and former head of the National Counterproliferation Center at the Office of the Director of National Intelligence. “Walking away from that capability? I don’t think there’s anything out there that says anything like that,” DeTrani says. Obama’s former top Iran adviser, Dennis Ross, wrote last month in the New York Times that Rouhani’s intentions were unclear, but that “it would be foolish to think that lifting the pressure now would improve the chances that he would be allowed to offer us what we need: an agreement, or credible Iranian steps toward one, under which Iran would comply with its international obligations on the nuclear issue.”

(MORE: Viewpoint: What Ancient Persian History Tells Us About Iran’s Future)

Ray Takeyh of the Council on Foreign Relations says the easy test is whether Rouhani agrees to the International Atomic Energy Agency requests for broader inspections of the Iranian nuclear program and other measures that Iran has been ignoring for years. The Administration’s likely approach, says Gary Samore, executive director of Harvard’s Belfer Center and Obama’s former top counterproliferation aide, is to stay the course, going neither softer nor harder on the Iranians. Samore says the U.S. will put the burden on Rouhani, awaiting his response to the latest diplomatic offer from Washington and the group of countries that have been working to reach a deal with Iran.

But if the U.S. is trying to put the ball in Iran’s court, Rouhani is already finding a way to volley it back to Washington. Asked by reporters Sunday if American and European officials had been invited to Rouhani’s Aug. 4 swearing-in at the Majlis, Iranian Foreign Ministry spokesman Abbas Araghchi said, “Officials of all countries have been invited.” The White House and the State Department did not respond immediately when asked if the U.S. would attend.

The stakes for Obama, and the U.S., are high in deciding how to respond to Rouhani. An agreement with Iran would be a lasting and important achievement for U.S. national security and for Obama’s presidency. But a successful Iranian effort to achieve nuclear status would be a potential disaster, launching either a new war in the Middle East, or a cascade of efforts by other countries in the region to get nuclear weapons too.

Update: Asked at his weekly press conference Tuesday which countries had been invited to Rouhani’s swearing in, Iranian Foreign Ministry spokesman Abbas Araghchi said, “Our invitation includes all countries, of course except the U.S. and the Zionist regime, which we essentially do not consider to be a country.” A senior Obama administration official confirmed that the U.S. did not receive an invitation.