Iran's delegation on Sunday walked out of the Swiss venue where it held talks with the United States towards permanently ending the Middle East war, after Donald Trump threatened to strike the Islamic republic over its support for Hezbollah.

The negotiations to end a war that sowed chaos across the region and rattled the global economy are meant to trigger a 60-day period to settle broader issues that have dogged US-Iranian relations for decades.

But disagreement over key sticking points and the threat of renewed fighting in Lebanon weighed on the talks, with Washington and Tehran exchanging threats in parallel with the negotiations.

"The delegation of the Islamic Republic of Iran, after meeting with the Qatari delegation as one of the mediating parties, left the building where the negotiations were being held," Iran's state news agency IRNA said.

"At the same time as the talks began in Switzerland, Donald Trump published a message on X in which he repeated his threats and remarks against Iran," it added.

A diplomat with knowledge of the talks, however, told AFP that the Iranians had not quit the negotiations.

"The Iranian delegation remains engaged in the talks and has not indicated to the mediators any intention to leave," the diplomat said on condition of anonymity.

Trump had earlier threatened to strike Iran if it did not "immediately stop their highly paid PROXIES in Lebanon from causing trouble".

Iran hit back with a warning of its own.

"They would do better to be careful with their statements; our armed forces are ready to respond to them in a different manner. No matter what they say, we are the ones who act," Iran's chief negotiator Mohammad Bagher Ghalibaf said. 

- 'New leaf?' -

Israeli Prime Minister Benjamin Netanyahu meanwhile said troops would remain in south Lebanon "as long as necessary" and vowed that he would "not allow Iran to acquire nuclear weapons".

Iran's state broadcaster said the nuclear programme had not been discussed "during the 80-minute first round of talks" on Sunday.

It said the focus had been on the implementation of the memorandum of understanding between Tehran and Washington, and the situation in Lebanon.

US Vice President JD Vance had earlier hailed "a historic meeting". 

Flanked by US negotiators Jared Kushner and Steve Witkoff in the luxury Swiss resort Burgenstock, Vance added: "The question before us now is how much more can we accomplish together?

"Can we turn over a new leaf? Can we change relations in the Middle East permanently?

"Or do we go back to doing things the old way, which is not our preference, but it's certainly very much something that can happen."

The talks to end the months-long conflict are taking place against the backdrop of Iran closing the strategic Strait of Hormuz again in response to recent Israeli attacks on Lebanon.

"It is not possible to enter the negotiation phase for a final agreement" unless there is an end to the war in Lebanon, Iranian foreign ministry spokesman Esmaeil Baqaei wrote on X.

Washington and Tehran's memorandum of understanding signed this week included a provision to end fighting in Lebanon between Israel and Hezbollah.

But there have been repeated clashes in Lebanon since, which prompted Iran to say it would again shutter the crucial oil and gas trade route, having opened it as part of the deal.

By Sunday evening, however, there had been no reports of Israeli strikes or continued fighting, with some residents of southern Lebanon cautiously trickling back to their homes.

- Sticking points  -

Lebanon aside, there has been no indication that Iran's support for armed groups across the region, which has long drawn the ire of the US and Israel, will be addressed in the negotiations.

Speaking on Sunday, Iranian President Masoud Pezeshkian said Tehran would not relinquish its right to enrich uranium, though he repeated Iran's denial that it wants nuclear weapons.

"We can also state in writing that we have no intention of building a bomb," he said.

The fighting between Israel and Hezbollah, sparked by the Tehran-backed militant group attacking Israel in support of Iran in the wider war, has repeatedly threatened to derail peace efforts.

On Friday, planned US-Iranian talks were postponed after Israel launched deadly strikes in Lebanon following the deaths of four of its soldiers in combat there.

Israel's military chief visited troops in south Lebanon on Sunday, where he said Hezbollah was in a "very difficult position".

"Hezbollah has suffered a severe and significant blow, and we are committed to remaining prepared to continue operating and prevent its rebuilding," Lieutenant General Eyal Zamir said.

The overall death toll from the fighting in Lebanon has now surpassed 4,100, the health ministry said.

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