Here is what we know about the behind-the-scenes mediation efforts:

- What is on the table? -

After announcing "very good" talks with an unnamed "top person" in Iran on Monday, Trump said Tuesday that he had sent a plan and that it "all starts with, they cannot have a nuclear weapon".

A 15-point proposal to stop the fighting was then conveyed to Iran via Pakistan, a key intermediary along with Egypt and Turkey, officials have confirmed.

But the contents are unknown and the identity of Trump's "top person" remains a mystery, if he exists at all.

The New York Times and Al Jazeera have reported that Trump is proposing a one-month ceasefire during which the two sides would restart talks about the same issues they were discussing before the war.

These include a US demand that Iran hand over its enriched uranium stockpile, stop any further enrichment, and agree to limits on its missile programme, as well as cease support for militant groups in the region.

If Iran met US conditions and opened up the strategic Strait of Hormuz for shipping, which it has effectively closed, Trump is offering relief from all sanctions, the reports suggest.

After saying that Iran wanted to make a deal "so badly", the US president warned Thursday that they "better get serious soon".

UN nuclear watchdog chief Rafael Grossi has said a meeting in Islamabad at the weekend is under discussion.

- What does Iran say? -

Publicly, Iranian officials insist no talks are taking place, but this means no direct dialogue has been established with Washington.

Foreign Minister Abbas Araghchi admitted Wednesday that messages were "being exchanged through friendly countries or through certain different individuals".

State-run Tasnim news agency, citing an unnamed official, reported Thursday that Iran had sent an official response to the United States on Wednesday evening.

"Iran's response to the 15 points proposed by the US was officially sent last night through intermediaries, and Iran is awaiting the other side's response," Tasnim quoted the source.

- What are Iran's demands? -

Unidentified Iranian officials, cited by Iran's state-run Press TV and Tasnim, have detailed five conditions for an end to hostilities.

These include ending "aggression and assassinations", setting up a mechanism guaranteeing that neither Israel nor the United States would resume the war, and financial compensation.

They also include a cessation of hostilities on all fronts, meaning Israel would stop bombarding Tehran-backed Hezbollah in Lebanon and possibly Hamas in Gaza.

The official also said Tehran wanted international recognition of its sovereignty over the Strait of Hormuz.

The Wall Street Journal reported this week that Tehran was also seeking the closure of US military bases in the Gulf.

- Can compromises be found? -

Analysts say the conflict has strengthened hardliners in Tehran where the rhetoric is defiant and increasingly confident.

Trump has bombed Iran twice amid negotiations, first in June last year and again on February 28.

As for US demands, Tehran has insisted since 2003 it is not seeking a nuclear weapon but has a right to enrich uranium for civilian nuclear energy purposes.

It has also refused curbs on its ballistic missile programme or discussions about its support for militant groups such as Hezbollah, or the Houthis in Yemen.

Tehran's demand for reparations is likely a non-starter, as is any suggestion of the US closing Gulf military bases.

It is not clear how its sovereignty over the Strait of Hormuz would work, or how meaningful security guarantees could be formulated, unless they involved outside powers such as Russia or China.

- So is there no hope? -

The outcome may come down to how badly Trump wants to end the war, and whether Iran's leaders view their interests as being best served by a ceasefire.

Trump would walk away claiming he had destroyed Iran's military and nuclear capabilities.

The Islamic republic could also claim victory, pointing to how its forces resisted four weeks of US and Israeli onslaught while also landing blows on US interests in the region and in Israel.

"Both sides need to be able to claim victory and save face, whatever deal they agree on," a diplomat based in the Middle East told AFP on condition of anonymity. "The process will take some time."

But even with a ceasefire, the question of Iran's nuclear programme and its 440-kilogram stockpile of highly enriched uranium remains unresolved.

Some analysts believe the talks are a smokescreen as Trump prepares a ground offensive to reopen the Strait of Hormuz by force or seize Iranian oil assets.

Iran has signalled it would use its Houthi allies in Yemen to attack shipping in the Red Sea, which would open up a new front in a war of spiralling economic, political and military repercussions.

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