Details of the original US proposal remain limited, but media outlets have reported it involves a one-page memorandum of understanding to end the fighting and set a framework for negotiations on Iran's nuclear programme.

President Donald Trump has flatly rejected Iran's response, leaving the push to end the Middle East war at a deadlock.

Here is what we know about the two sides' positions:

- Nuclear question -

Iran's response called for an immediate end to the war across the region, including in Lebanon, as well as a halt to the US naval blockade on Iranian ports, according to the Iranian foreign ministry on Monday.

It also called for the release of Iranian assets frozen abroad under longstanding US sanctions, the ministry said.

According to anonymous sources cited by the Wall Street Journal, Iran proposed engaging in negotiations on its nuclear programme over "the next 30 days".

It also offered "to have some of its highly enriched uranium diluted and the rest transferred to a third country", the newspaper said.

The Journal reported that Iran was willing to suspend enrichment of uranium but for a shorter period than 20 years proposed by the US. But Tehran rejected dismantling its nuclear facilities, the outlet said.

Israel and the United States bombed Iranian nuclear sites during a 12-day war last June and Trump has claimed the strikes "obliterated" the nuclear programme.

But the two countries say Iran still possesses enriched uranium that could be used to manufacture a bomb, saying it must be removed from Iran.

The Journal said Iran's response had failed to meet US demands on the future of the nuclear programme and highly enriched uranium.

Trump on Sunday called the Iranian reply "TOTALLY UNACCEPTABLE".

Iran's Tasnim news agency, citing an unnamed source, said the paper's report "was not reflective of the reality", particularly regarding the highly enriched uranium.

- Hormuz reopening? -

Iran's response also proposed a gradual reopening of the Strait of Hormuz to commercial shipping, the Journal reported, after Iran blocked maritime traffic through the route following the outbreak of war.

In return, Iran proposed the US implement a phased easing of its naval blockade on Iranian ports that Washington began on April 13.

Iran has been setting up a payment mechanism to charge tolls for ships crossing the strait, through which a fifth of the world's oil and liquefied natural gas flow during peacetime.

The restrictions have sent prices of key commodities soaring and roiled the global economy.

Last month Washington retaliated by imposing its own blockade on Iranian shipping, aiming to choke off the country's oil exports.

According to Tasnim, the Iranian response focuses on lifting sanctions on Iranian oil sales and releasing Iranian assets frozen overseas within a 30-day period.

But Tasnim did not offer details on what Iran would offer in return. It said Iran's response had emphasised that Tehran would continue to manage Hormuz.