The United States said that it was sending additional troops to the Middle East, after protesters stormed its embassy compound in Iraq, setting fires and chanting: "Death to America!"

Angered by US air raids that killed two dozen paramilitary fighters on Sunday, hundreds of men spilled through checkpoints in the high-security Green Zone on Tuesday, demanding the removal of American troops from Iraq.

Reacting to the attack, US President Donald Trump said he held Tehran "fully responsible" for the incident, and said the protesters "will be held fully responsible", he wrote on Twitter on Tuesday.

Trump later warned in a separate statement that Tehran would "pay a very big price" after the attack, but when asked later in the day about the possibility of tensions spiralling into a war with Iran, Trump told reporters: "Do I want to? No. I want to have peace. I like peace. And Iran should want to have peace more than anybody. So I don't see that happening."

US Secretary of State Mike Pompeo said the attack was "orchestrated by terrorists", one of whom he named as Abu Mahdi al-Muhandis.

Al-Muhandis has been identified as the second-in-command of the Tehran-backed Hashd al-Shaabi (Popular Mobilization Forces) paramilitary group which includes Kataib Hezbollah, the group that was hit in the US air raids.

Defense Secretary Mark Esper said in a statement that about 750 troops from a rapid response unit of the 82nd Airborne Division are prepared to deploy over the next several days to the region.

A social media post from the US Marines said the troops from its crisis response command in Kuwait were deployed to Iraq.

Hundreds of protesters stormed the US embassy compound in Baghdad's heavily fortified Green Zone/Photo: Khalid Mohammed

The deployment of 750 soldiers is in addition to 14,000 US troops who have been deployed to the Gulf region since May in response to concerns about Iran, and the reported attacks on commercial shipping in the Gulf.

At the time of the attack, the US had about 5,200 troops in Iraq, mainly to train Iraqi forces and help them combat the ISIL (ISIS) group.

Meanwhile, a US official, who provided unreleased details to the Associated Press news agency on condition of anonymity, said the full brigade of about 4,000 soldiers may deploy.

On Sunday, the US launched air attacks on sites in Iraq and Syria belonging to Kataib Hezbollah, an Iran-backed fighter group, that Washington said was in retaliation for the killing of a US contractor.

At least 25 fighters of Kataib Hezbollah were killed in the attack.

Kataib Hezbollah, through its spokesman Mohammed Muhi, denied carrying out the attack on the US facility.