WASHINGTON (AFP) -- President Donald Trump warned Saturday that the US is targeting 52 sites in Iran and will hit them "very fast and very hard" if the Islamic republic attacks American personnel or assets.

In a saber-rattling tweet that defended Friday's US drone strike assassination of a powerful Iranian general in Iraq, Trump said 52 represents the number of Americans held hostage at the US embassy in Tehran for more than a year starting in late 1979.

Trump said some of these sites are "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 spoke out after pro-Iran factions ramped up pressure on US installations across Iraq with missiles and warnings to Iraqi troops -- part of an outburst of fury over the killing of Qasem Soleimani, described as the second most-powerful man in Iran.

With Iran promising revenge, his killing was the most dramatic escalation yet in spiraling tensions between Washington and Tehran and has prompted fears of a major conflagration in the Middle East.

In the first hints of a possible retaliatory response, two mortar rounds hit an area near the US embassy in Baghdad on Saturday, security sources told AFP.

Almost simultaneously, two rockets slammed into the Al-Balad airbase where American troops are deployed, security sources said.

The Iraqi military confirmed the missile attacks in Baghdad and on al-Balad and said there were no casualties. The US military also said no coalition troops were hurt.

With Americans wondering fearfully if, how and where Iran will hit back for the assassination, the US Department of Homeland Security issued a bulletin that said "at this time there is no specific, credible threat against the homeland."

While no one claimed the attacks in Baghdad, a hardline pro-Iran faction in Iraq's Hashed al-Shaabi military network shortly after urged Iraqis to move away from US forces.

"We ask security forces in the country to get at least 1,000 meters away from US bases starting on Sunday at 5:00pm (1400 GMT)," said Kataeb Hezbollah.

The deadline would coincide with a parliament session on Sunday which the Hashed has insisted should see a vote on the ouster of US troops. 

Washington has blamed the vehemently anti-American group for a series of rocket attacks in recent weeks targeting US diplomats and troops stationed across Iraq.

- 'Direct war' -

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Many fear the American strike that killed Iran's military mastermind Soleimani would set off a wider conflict with Iran, and have braced for more attacks.

"This is no longer a proxy war," said Erica Gaston, a non-resident fellow at the New America Foundation.

"What you have is America attacking an Iranian general directly, and groups are now openly fighting for Iran to avenge him. This is a direct war," she told AFP.

The US strike on Baghdad international airport early Friday killed a total of five Iranian Revolutionary Guards and five members of Iraq's Hashed.

Among the dead was Hashed's deputy head Abu Mahdi al-Muhandis, a top adviser and personal friend to Soleimani.

As head of the Guards' foreign operations arm, the Quds Force, Soleimani was a powerful figure domestically and oversaw Iran's wide-ranging interventions in regional power struggles.

Trump has said Soleimani was planning an "imminent" attack on US personnel in Baghdad and should have been killed "many years ago".

Iran's supreme leader Ayatollah Ali Khamenei promised "severe revenge" for Soleimani's death and Tehran named Soleimani's deputy, Esmail Qaani, to succeed him.

Tens of thousands of Iraqis including Prime Minister Adel Abdel Mahdi, political leaders and clerics attended a mass ceremony on Saturday to honor Soleimani and the other victims.

They waved white Hashed flags and massive portraits of Iranian and Iraqi leaders, furiously calling for "revenge" and chanting "Death to America!"

The remains were moved from Baghdad to the shrine city of Karbala and then Najaf, where the Iraqis will be buried and from whose airports the Guards are to be flown to Iran.

Tehran has slammed the strike as an "act of war" and Abdel Mahdi said it could bring "devastating" violence to Iraq.

World powers quickly called for a de-escalation.

Iranian Foreign Minister Mohammad Javad Zarif mocked as "foolish" a diplomatic effort by the United States, which he said had sent a letter to Iranian officials through a Swiss envoy since Tehran and Washington have not had direct diplomatic ties for decades.

- Ousting 'the occupier' -

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The attacks on Saturday evening appeared to be precisely the reaction Iraqis had long feared: tit-for-tat strikes between the Hashed and the US on Iraqi soil.

Early on Saturday, the Hashed had claimed a new strike hit their convoy north of Baghdad, with Iraqi state media blaming the US.

But the US-led coalition denied involvement, telling AFP: "There was no American or coalition strike" on Saturday.

Iraq's pro-Iran factions have seized on Soleimani's death to demand parliament decree that US forces leave Iraq.

"We either vote on the occupation forces leaving, or we remain subservient, robbed of our will and dignity," said MP Ahmad al-Kinany of the Hashed's political bloc, Fatah. (AFP)