Five weeks into the Middle East war triggered by a joint US-Israeli air assault on Tehran, the US leader has demanded that Iran reopen the Strait of Hormuz to international shipping by midnight GMT on Tuesday, or face a newly devastating round of bombing.

Both Trump and Iran have said that a proposal touted by international mediators for a 45-day ceasefire is not yet ready, and in a Washington press conference, the US president dialled up his warlike rhetoric once again.

"We have a plan -- because of the power of our military -- where every bridge in Iran will be decimated by 12 o'clock tomorrow night, where every power plant in Iran will be out of business, burning, exploding and never to be used again," Trump said.

"I mean complete demolition by 12 o'clock, and it'll happen over a period of four hours -- if we wanted to."

Trump had earlier accepted the ceasefire plan was a "significant proposal", but went on to say it was not good enough. Iranian state media quotes officials stating that Tehran too "has rejected a ceasefire and insists on the need for a definitive end to the conflict".

Trump said intermediaries "are negotiating now" on improving the ceasefire proposal, which US media reported was being mediated by Pakistan, Egypt and Turkey.

Iran's military said it would "continue the war as long as the political authorities see fit".

Trump's latest threats, including a profanity-laced social media post on Sunday, have sent shockwaves through the international community.

International Committee of the Red Cross chief Mirjana Spoljaric warned that "deliberate threats... against essential civilian infrastructure" are illegal.

But talk of a ceasefire came as the US and Israel were striking targets across Iran, including major petrochemical facilities, and as Iran continued missile and drone attacks around the region.

Iran's virtual blockade of Hormuz has sent oil and gas prices soaring and pushed countries around the world to enact measures to contain the fallout.

- 'We will reach anyone' -

Earlier Monday, Israeli strikes had hit major Iranian petrochemical facilities including in Asaluyeh on the Gulf coast, the country's biggest, and another outside Shiraz in central Iran.

Israel's military said it had also struck Iranian air force targets including planes and helicopters at airports in Tehran and elsewhere.

Iran's Guards posted on Telegram on Monday that their intelligence chief Majid Khademi had been killed at dawn in US-Israeli strikes.

Israel's military also said it had killed Asghar Bagheri, commander of the Guards' Quds Force special operations unit, on Sunday.

"We will reach anyone who seeks to harm us," Israeli Prime Minister Benjamin Netanyahu said.

The Guards' Intelligence Organisation vowed a "major retaliatory strike" against those responsible for killing their commanders, their official Sepah News website reported.

Yemen's Houthi rebels said they launched an attack targeting Israel, supporting their backer Iran and Lebanon's Hezbollah.

- Oil squeeze -

The war, which erupted on February 28 with US-Israeli strikes on Iran that killed supreme leader Ayatollah Ali Khamenei, has engulfed the Middle East and roiled the global economy.

The worldwide oil squeeze has hit aviation, with Indonesia on Monday saying it would increase a jet fuel surcharge and low-cost carrier Air Asia X announcing ticket price hikes of up to 40 percent.

South Korea will send ships to fetch oil from Saudi Arabia's Red Sea port of Yanbu, avoiding Hormuz altogether, a ruling party MP said, while Taiwan's government said it too would take the Red Sea route.

Gulf nations allied with the US have also been sucked into the war, with Kuwait and the UAE reporting strikes and injuries from Sunday to Monday.

Iran has continued to launch attacks at Israel, where the military and medics said four bodies were recovered from a residential building in the northern city of Haifa that was struck by a missile.

Iranian media reported several attacks on residential areas of Tehran, while the state broadcaster said gas outages hit parts of the capital after a strike on a university.

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