The Pentagon said US President Donald Trump ordered Soleimani's "killing," after a pro-Iran mob this week laid siege to the US embassy.

Following Soleimani's death, Trump tweeted an image of the US flag without any further explanation.

The pre-dawn developments mark the most major escalation yet in a feared proxy war between Iran and the US on Iraqi soil.

Early Friday, a volley of missiles hit Baghdad's international airport, striking a convoy belonging to the Hashed al-Shaabi, an Iraqi paramilitary force with close ties to Iran.

Just a few hours later, Iranian state television carried a statement by the Revolutionary Guard Corps saying Soleimani "was martyred in an attack by America on Baghdad airport this morning."

The Hashed confirmed that both Soleimani and its deputy chief Abu Mahdi al-Muhandis were killed in what it said was a "US strike."

"The deputy head of the Hashed, Abu Mahdi al-Muhandis, and head of the Quds Force, Qasem Soleimani, were killed in a US strike that targeted their car on the Baghdad International Airport road," it said.

The Hashed is a network of mostly-Shiite armed units, many of whom have close ties to Tehran but which have been officially incorporated into Iraq's state security forces.

The units joined forces to fight the Islamic State group in 2014, after many of them built up years of fighting experience during Iraq's war years, including against the US.

- 'Major decapitation strike' -

Muhandis was the Hashed's deputy chief but widely recognised as the real shot-caller within the group.

Soleimani headed the Islamic Revolutionary Guard Corps' Quds Force and also served as Iran's pointman on Iraq, visiting the country in times of turmoil.

Both were sanctioned by the United States.

The Pentagon said Soleimani had been "actively developing plans to attack American diplomats and service members in Iraq and throughout the region."

It said it took "decisive defensive action to protect US personnel abroad by killing Qasem Soleimani," but did not specify how.

Phillip Smyth, a US-based specialist in Shiite armed groups, told AFP the strike would have "bigger" ramifications than the 2011 US operation that killed Al-Qaeda chief Osama bin Laden and the 2019 American raid that killed Islamic State group Abu Bakr al-Baghdadi.

"In terms of a decapitation strike, what just happened is the most major decapitation strike that the US has ever pulled off," said Smyth.

"There is no comparison," he added.

The developments come after an unprecedented attack on the US mission in Baghdad.

A mob of Hashed supporters surrounded the US embassy on Tuesday in outrage over American air strikes that killed 25 fighters from the network's hardline Kataeb Hezbollah faction, which is backed by Iran.

The US had acted in response to a rocket attack days earlier that had killed an American contractor working in Iraq.

Trump had blamed Iran for a spate of rocket attacks targeting US forces as well as the siege at the embassy, saying: "They will pay a very BIG PRICE! This is not a Warning, it is a Threat."

On Thursday, US Secretary of Defence Mark Esper told reporters there could be more attacks from Iran-backed groups but vowed American forces would respond.

"We are prepared to exercise self-defense, and we are prepared to deter further bad behavior from these groups, all of which are sponsored, directed and resourced by Iran," he said.

- Oil prices soar -

Ties between the US and Iran have deteriorated since Washington pulled out of the landmark nuclear deal with Tehran in 2018.

It then reimposed crippling sanctions on Iran, aiming to choke off its oil exports.

Oil prices soared following Friday's attack, with Brent surging 4.4 percent to $69.16 and WTI jumping 4.3 percent to 63.84.

But the attack threatens to destabilise Iraq, which has close political and military ties with both Tehran and Washington.

The United States led the 2003 invasion against then-dictator Saddam Hussein and has worked closely with Iraqi officials since.

But its influence has waned compared with that of Tehran, which has carefully crafted personal ties with Iraqi politicians and armed factions, even during Saddam's reign.

Iraqi officials have warned in recent months that their country could be used as an arena for score-settling between Iran and the US.