* US-Iran ties cut in 1980 after hostage crisis
* Rouhani has signalled desire to end nuclear dispute
* He and Obama to address UN on same day next week
DUBAI, Sept 17 (Reuters) - Iran said on Tuesday that President Hassan Rouhani had exchanged letters with U.S. President Barack Obama, confirming a rare contact between leaders of the two nations at loggerheads over Iran's nuclear programme, the Syrian war and other issues.
The United States and Iran cut diplomatic relations in 1980, after students and Islamic militants stormed the U.S. Embassy in Tehran and took American diplomats hostage.
But since the surprise election of Rouhani, a centrist cleric who defeated more conservative candidates, in June, officials from both countries have made increasing hints that they are open to direct talks to seek an end to the decade-long nuclear dispute.
The United States and Europe have imposed sanctions on Iran's economy, including its vital oil sector, over concerns it is working towards nuclear weapons capability. Tehran denies that and says the nuclear issue is being used as an excuse to punish a country unpopular in the West.
Iranian Foreign Ministry spokeswoman Marzieh Afkham said on Tuesday that Obama had sent Rouhani a message of congratulations on the occasion of his election.
"This letter has been exchanged," Afkham said, according to the ISNA news agency. "The mechanism for exchanging these letters is through current diplomatic channels."
Though rare, it is not the first time letters have been exchanged. Rouhani's predecessor, Mahmoud Ahmadinejad, wrote one to Obama three years ago, and Obama wrote twice directly to Supreme Leader Ayatollah Ali Khamenei, in 2009 and 2012.
Obama said in an interview broadcast on Sunday he had exchanged letters with Rouhani. The two men will speak on the same day at the U.N. General Assembly next week, though there are currently no plans for them to meet.
Rouhani has said he wants to pursue "constructive interaction" with the world, raising hopes for a negotiated settlement to the nuclear dispute.
Another indication came on Tuesday from Supreme Leader Ayatollah Ali Khamenei, who would have to authorise any nuclear deal. In a speech, he said he supported "flexibility" when it came to Iran's diplomacy, though he did not say what that might mean in practice.
Khamenei also said he supported "correct and rational foreign and domestic policies," but warned that Iran should not forget that it had enemies.
At a U.N. nuclear agency meeting in Vienna on Tuesday, the head of the French delegation, Bernard Bigot, noted Iran's stated intention to dispel concerns about its nuclear programme.
"We would like to take this assurance in good faith, but for us to do so Iran must act as soon as possible to take concrete, verifiable measures," he said.