Security in the Middle East expected to dominate agenda as Trump and Rouhani address annual gathering of world leaders.
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The United States and Iran are set to put forward their competing visions of security in the Middle East to the United Nations General Assembly this week, with US President Donald Trump expected to address the gathering of world leaders on Tuesday, and Iranian President Hassan Rouhani the following day.
Even as tensions continue to simmer in the Middle East, following the attack on Saudi Arabia's oil fields and the exchange of sharp rhetoric between Washington and Tehran, the two leaders are holding out hope for diplomacy, with Trump saying he is "a very flexible person", and Rouhani extending a "hand of friendship and brotherhood" towards its regional neighbours.
On Sunday, Trump once again left open the possibility of an unscheduled meeting with Rouhani on the sidelines of the UN General Assembly.
"Nothing is ever off the table, completely, but I have no intention of meeting with Iran and that doesn't mean it doesn't happen," Trump said. "I'm a very flexible person, but we have no intention. It's not set up."
When asked if there would be any discussions between US Secretary of State Mike Pompeo or another American representative with the Iranian delegation at the UN General Assembly, a high-ranking American official said there was "nothing planned."
But CNN's Christiane Amanpour tweeted that Iranian Foreign Minister Javad Zarif told her in an interview that Rouhani was willing to meet Trump in New York this week "provided that President Trump is ready to do what's necessary" by exchanging sanctions relief for "permanent monitoring of Iranian nuclear facilities."
"The olive branch has always been on the table, but we're showing it again," Zarif added, according to Amanpour's tweet.
In a televised speech earlier on Sunday marking the anniversary of the start of the Iran-Iraq war in 1980, Rouhani said that Iran was extending a "hand of friendship and brotherhood" to neighbouring countries to secure the Gulf and the Strait of Hormuz - a vital gateway for the global oil industry.
But the Iranian president also warned foreign forces to "stay away" from the region.
"Foreign forces can cause problems and insecurity for our people and for our region," Rouhani said.
'Risks of miscalculation'
Meanwhile, in an interview with the US broadcaster ABC on Sunday, Pompeo, the top US diplomat, said Washington wants to give diplomacy "every opportunity to succeed".
Pompeo put aside threats of US military retaliation against Iran, saying Trump's administration is "taking this on in a serious way and we are working diligently to see that this has a diplomatic outcome."
"But make no mistake about it, if we're unsuccessful in that and Iran continues to strike out in this way, I am confident that President Trump will make the decisions necessary to achieve our objectives."
The United States has accused Tehran of carrying out the air attacks that set aflame Saudi Arabia's Abqaiq plant and the Khurais oil field on September 14, knocking out half the kingdom's oil production.
Pompeo called it "one of the largest attacks on the global energy supply in history."
But despite initial warnings by Trump that US forces were "locked and loaded," the US president quickly softened his rhetoric, brushing off Republican hawks who warned that the absence of a forceful response would be read as weakness in Tehran.
Trump has so far ordered stepped up sanctions against Iran, and a relatively modest deployment of US troops to the Gulf, primarily focused on defensive missions like air and missile defense.
On Fox News, Pompeo said the administration was "deeply aware of the risks" of a miscalculation given the situation in the region.
"It's why we want to resolve this in a way that doesn't resort to kinetic action if it's at all possible to achieve that," he said.
The threat of outright war is likely to make all sides cautious.
"I doubt anyone has the appetite for a direct clash between the US and Iran," said Barbara A. Leaf, who was US ambassador to the UAE from 2014-2018 and is now a senior fellow at the Washington Institute for Near East Policy.
"It really requires re-establishing deterrence. Clearly there is none now."
Meanwhile, Mohammad Eslami, a former editor-in-chief of Iran's Khorasan diplomatic magazine, said he doubts that Trump and Rouhani would meet in New York.
"President Trump is not reliable in the way Rouhani thinks about diplomacy," Eslami told Al Jazeera, adding that the US president is only interested in showing off that he is doing something on Iran before the 2020 elections.
Eslami also said that meeting Trump would be "political suicide for any Iranian official".
"The cultural factor and the history of Iran-US relations are the main contexts when we are writing about diplomacy. Trump does not respect the Iranian heritage and the Iranian culture," he said.