The United States said on Tuesday 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 that the protesters "will be held fully responsible," he wrote on Twitter on Tuesday.

"In addition, we expect Iraq to use its forces to protect the Embassy, and so notified!"

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 spiraling 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 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.

"This deployment is an appropriate and precautionary action taken in response to increased threat levels against US personnel and facilities, such as we witnessed in Baghdad today," he said.

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

The 750 soldiers deploying immediately were in addition to 14,000 US troops who had 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 ISIL (Islamic State of Iraq and the Levant, also known as ISIS).

Meanwhile, a US official, who provided unreleased details to the Associated Press 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 militia, that Washington said was in retaliation for the killing of a US contractor on an Iraqi military base.

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

Kataib Hezbollah, through its spokesman Mohammed Muhi, denied responsibility for the attack on the US facility.

That triggered the protest and storming of the US embassy compound on Tuesday.

The protesters raised flags of the powerful paramilitary group Hashd al-Shaabi (Popular Mobilization Forces), with which Kataib Hezbollah is associated.

Downturn in Iraq-US relations

The developments represent a significant downturn in Iraq-US relations that could further undermine US influence in the region and weaken Washington's hand in its pressure campaign against Iran.

Iraq has long struggled to balance its ties with the US and Iran, allies of the Iraqi government.

But the government's angry reaction to the US air attacks signalled a deterioration of US-Iraq relations.