US President Donald Trump on Saturday threatened to hit 52 Iranian sites "very hard" if Iran attacks Americans or US assets after a drone strike that killed Iranian military commander Qasem Soleimani and an Iraqi militia leader, as tens of thousands of people marched in Iraq to mourn their deaths.

Showing no signs of seeking to ease tensions raised by the strike he ordered that killed Soleimani and Iranian-backed Iraqi militia leader Abu Mahdi al-Muhandis at Baghdad airport on Friday, Trump issued a threat to Iran on Twitter. The strike has raised the spectre of wider conflict in the Middle East.

Iran, Trump wrote, "is talking very boldly about targeting certain USA assets" in revenge for Soleimani's death.

Trump said the United States has "targeted 52 Iranian sites" and that some were "at a very high level & important to Iran & the Iranian culture, and those targets, and Iran itself, WILL BE HIT VERY FAST AND VERY HARD."

"The USA wants no more threats!" Trump said, adding that the 52 targets represented the 52 Americans who were held hostage in Iran for 444 days after being seized at the US Embassy in Tehran in November 1979 — an enduring sore spot in US-Iranian relations.

Trump did not identify the sites. The Pentagon referred questions about the matter to the White House, which did not immediately respond to a request for comment.

In a subsequent tweet, Trump warned that the United States would hit Iran harder than ever before if Tehran retaliates to the assassination of one of its top generals.

He tweeted: "If they attack again, which I would strongly advise them not to do, we will hit them harder than they have ever been hit before!"

Trump followed up with another tweet, saying the US would use its "brand new beautiful" military equipment "without hesitation" if the Iranians retaliate.

Hours after Trump's threats Iran's top diplomat said any decision to target the country's cultural sites would be a "war crime".

"Targeting cultural sites is a WAR CRIME," Mohammad Javad Zarif tweeted in response to a post by Trump.

Among the mourners in Iraq included many militiamen in uniform for whom Muhandis and Soleimani were heroes.

They carried portraits of both men and plastered them on walls and armoured personnel carriers in the procession. Chants of "Death to America" and "No No Israel" rang out.

On Saturday evening, a rocket fell inside Baghdad's heavily fortified Green Zone near the US Embassy, another hit the nearby Jadriya neighbourhood and two more were fired at the Balad air base north of the city, but no one was killed, Iraq's military said. There was no immediate claim of responsibility.

Trump referenced an unusually specific number of potential Iranian targets after a senior Iranian Revolutionary Guards commander had also mentioned a specific number of American targets — 35 of them — for possible retaliatory attacks in response to Soleimani's killing.

General Gholamali Abuhamzeh was quoted by Tasnim news agency as saying late on Friday that Iran will punish Americans wherever they are within reach of the Islamic Republic, and raised the prospect of attacks on ships in the Gulf.

"The Strait of Hormuz is a vital point for the West and a large number of American destroyers and warships cross there. [...] Vital American targets in the region have been identified by Iran since long time ago. [...] Some 35 US targets in the region as well as Tel Aviv are within our reach," he was quoted as saying.

Iraq's Kataib Hezbollah militia warned Iraqi security forces to stay away from US bases in Iraq, "by a distance not less than a thousand metres (six-tenths of a mile) starting Sunday evening," reported Lebanese al-Mayadeen TV, which is close to Lebanon's Hezbollah.

Trump said on Friday Soleimani had been plotting "imminent and sinister" attacks on American diplomats and military personnel. Democratic critics said the Republican president's action was reckless and risked more bloodshed in a dangerous region.

'Malign influence'

Trump's provocative Twitter posts came only hours after U.S. Secretary of State Mike Pompeo wrote on Twitter that he had told Iraq's president that "the US remains committed to de-escalation".

Pompeo also wrote on Twitter that he had spoken with Israeli Prime Minister Benjamin Netanyahu about Iran and "underscored the importance of countering Iran's malign influence and threats to the region".

With security worries rising after Friday's strike, the Nato alliance and a separate US-led mission suspended their programmes to train Iraqi security and armed forces, officials said.

"The safety of our personnel in Iraq is paramount. We continue to take all precautions necessary," acting Nato spokesman Dylan White said in a statement.

Soleimani, 62, was Iran's pre-eminent military leader — head of the Revolutionary Guards' overseas Quds Force and the architect of Iran's spreading influence in the Middle East. Muhandis was de facto leader of Iraq's Popular Mobilisation Forces (PMF) umbrella body of paramilitary groups.

The attack took Washington and its allies, mainly Saudi Arabia and Israel, into uncharted territory in their confrontation with Iran and its proxy militias across the region.

France stepped up diplomatic initiatives on Saturday to ease tensions. French President Emmanuel Macron talked with Iraq President Barham Salih, Macron's office said. Macron also spoke with the de facto ruler of the United Arab Emirates, Abu Dhabi Crown Prince Sheikh Mohammed bin Zayed al-Nahyan.

The United States has been an ally of the Iraqi government since the 2003 US invasion to oust dictator Saddam Hussein, but Iraq has become more closely allied with Iran.

The Iraqi parliament is convening an extraordinary session during which a vote to expel US troops could be taken as soon as Sunday. Many Iraqis, including opponents of Soleimani, have expressed anger at Washington for killing the two men on Iraqi soil and possibly dragging their country into another conflict.

Bodies take to holy cities

A PMF-organised procession carried the bodies of Soleimani and Muhandis, and those of others killed in the US strike, through Baghdad's Green Zone.

The top candidate to succeed Muhandis, Hadi al-Amiri, spoke over the dead militia commander's coffin: "The price for your noble blood is American forces leaving Iraq forever and achieving total national sovereignty."

Prime Minister Adel Abdul Mahdi also attended. Mahdi's office later said he received a phone call from Saudi Crown Prince Mohammed bin Salman and they "discussed the difficult conditions facing Iraq and the region".

Mourners brought the bodies of the two slain men by car to the Shia holy city of Karbala, south of Baghdad, then to Najaf, another sacred Shia city, where they were met by the son of Iraq's top Shia cleric, Grand Ayatollah Ali al-Sistani, and where Muhandis and the other Iraqis killed will be laid to rest.

Soleimani's body will be transferred to the southwestern Iranian province of Khuzestan that borders Iraq. On Sunday it will be taken to the Shia holy city of Mashhad in Iran's northeast and from there to Tehran and his hometown Kerman in the southeast for burial on Tuesday, state media said.

The US strike followed a sharp increase in US-Iranian hostilities in Iraq since last week when pro-Iranian militias attacked the US embassy in Baghdad after a deadly US air raid on Kataib Hezbollah, founded by Muhandis. Washington accused the group of an attack on an Iraqi military base that killed an American contractor.