Iraq's political reform process was in limbo Monday after protesters demanding a change of government reacted to weeks of stalling by storming parliament.

Security concerns were also high due to the presence in Baghdad of thousands of Shiite pilgrims, who were targeted for the second time in three days with a suicide bombing that killed at least 14 people.

Demonstrators pulled out of the Green Zone, where parliament is located, on Sunday evening, a day after breaching the walls of the fortified government district.

But the protesters, most of whom are followers of outspoken cleric Moqtada al-Sadr, also warned they would be back on the streets of Baghdad on Friday if their demands were not met.

Sadr supports Prime Minister Haider al-Abadi's efforts to form a new cabinet of technocrats to replace the current government of party-affiliated ministers, accused of graft and sectarianism.

There was no clear plan of action emerging Monday from any of the main players and several sources said Sadr himself flew to neighbouring Iran, the main foreign broker among Shiite political blocs in Iraq.

"The leader of the Sadrist movement left at 11:00 am from Najaf airport to the Imam Khomeini airport" southwest of Tehran, a Najaf airport official told AFP.

"Sadr took two other clerics with him," the official said, speaking on condition of anonymity.

A political source in Baghdad had the same information but there was no confirmation from either his office or Tehran.

Iraq's lawmakers looked unlikely to hold another session this week however, with the main parliament building requiring a massive cleanup following Saturday's events.

Thousands of mostly Sadrist protesters pulled down blast walls around the Green Zone and stormed the chamber after MPs again failed to agree on reforms.

Some MPs were roughed up on Saturday and their vehicles vandalised, and lawmakers appeared wary of exposing themselves to another attack.

"It was decided to hold a parliamentary session next week in another place because the (parliament) hall was damaged," MP Abbas al-Bayati told AFP.

There was no official statement from the speaker on the issue however.

- IS claims fresh bombing -

Abadi called for those who committed violent acts on Saturday to be arrested, but his grip on Iraq's top job looked more tenuous than ever.

A senior official in the Dawa party, of which Abadi is a member, said there was discussion within the party of the premier's resignation.

"We are in a debate inside the party for the first time (on) the demand for Abadi to resign," the official said.

A Sunni MP, Ahmed al-Masari, said his bloc was demanding the restructuring of the political process through... meetings in the coming days".

Another lawmaker said Kurdish MPs in the federal parliament were weighing their participation altogether.

Since coming to power in September 2014, Abadi has faced tough opposition from his predecessor and fellow party member Nuri al-Maliki.

Abadi nonetheless enjoys the support of Western powers, who have warned that continued political deadlock risks hampering Iraq's fight against the Islamic State group, which seized control of large parts of the country in mid-2014.

Backed by a US-led coalition, Iraqi security forces have made significant gains in retaking territory from IS in recent months, but still face huge challenges in rooting out jihadist fighters from the western province of Anbar and the country's second city of Mosul.

As the "caliphate" the jihadists proclaimed nearly two years ago continues to shrink, they have increasingly reverted to targeting civilians in bombings in Iraq's cities.

IS claimed responsibility for Monday's bombing that left at least 14 dead, saying one of its suicide bombers had detonated a car bomb against Shiite pilgrims in southern Baghdad.

They were walking to the northern Baghdad shrine of Imam Musa Kadhim, whose death in 799 AD is an important date in the Shiite Muslim calendar and is commemorated annually.

Another 23 people were killed in a similar attack on the outskirts of the Iraqi capital on Saturday.

The religious commemoration is due to culminate on Tuesday with tens of thousands of faithful converging on the shrine in Baghdad's Kadhimiya neighbourhood.