Supporters of the Hashed al-Shaabi paramilitary force attend the funeral procession of Iraqi paramilitary chief Abu Mahdi al-Muhandis, Iranian military commander Qasem Soleimani, and eight others in the capital Baghdad's district of al-Jadriya, near the high-security Green Zone, on January 4, 2020.

With shouts of "Death to America," tens of thousands of people marched in Iraq on Saturday to mourn Iranian military commander Qassem Soleimani and an Iraqi militia leader who were killed in a U.S. air strike that has raised the specter of wider conflict in the Middle East.

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

With security worries rising after Friday's strike, the NATO alliance and a separate U.S.-led mission suspended their programs 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, commander of the Revolutionary Guards' foreign legions, was killed in the U.S. strike on his convoy at Baghdad airport. The attack took Washington and its allies, mainly Saudi Arabia and Israel, into uncharted territory in its confrontation with Iran and its proxy militias across the region.

France stepped up diplomatic initiatives on Saturday to ease tensions in the Middle East. French President Emmanuel Macron talked with Iraq President Barham Salih, Macron's office said.

"The two presidents agreed to remain in close contact to avoid any further escalation in tensions and in order to act to ensure stability in Iraq and the broader region," a statement from Macron's office said.

Macron also discussed the situation with the de facto ruler of the United Arab Emirates, Abu Dhabi Crown Prince Sheikh Mohammed bin Zayed al-Nahyan.

Gholamali Abuhamzeh, a senior commander of the Revolutionary Guards, said Tehran would punish Americans "wherever they are in reach", and raised the prospect of possible attacks on ships in the Gulf.

The U.S. Embassy in Baghdad urged American citizens to leave Iraq. Dozens of American employees of foreign oil companies left the southern Iraqi city of Basra on Friday.

Close U.S. ally Britain warned its nationals to avoid all travel to Iraq, outside the autonomous Kurdistan region, and to avoid all but essential travel to Iran.

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.

The Iraqi militia leader killed in the strike, Abu Mahdi al-Muhandis, was the deputy commander of Iraq's Popular Mobilisation Forces (PMF) umbrella body of paramilitary groups.

A PMF-organised procession carrying the bodies of Soleimani, Muhandis and other Iraqis killed in the U.S. strike took place in Baghdad's Green Zone.

Mourners included many militiamen in uniform for whom Muhandis and Soleimani were heroes. They carried portraits of both men and plastered them on walls and armored personnel carriers in the procession. Chants of "Death to America" and "No No Israel" rang out.