Iraqi Prime Minister Nouri al-Maliki urged residents of Fallujah to "expel" al-Qaida fighters who have occupied the city centre.

The Sunni militants overran Fallujah and Anbar provincial capital Ramadi last week.

Mr Maliki also urged government troops to avoid targeting residential areas, but he did not say how he expected residents and pro-government tribes to push the militants out.

Government troops have reportedly surrounded Fallujah.

Fighting in majority-Sunni Anbar province killed at least 22 soldiers, 12 civilians and an unknown number of militants on Sunday as government troops tried to dislodge the al-Qaida-linked Islamic State in Iraq and the Levant (Isil).

Dozens of families fled the cities as clashes continued into the early hours.

Sectarian violence flared up in the wake of the disastrous US-led invasion of Iraq and has only intensified since foreign troops pulled out.

The Shi'ite-led government has been accused of discrimination against the Sunni minority, who have been further irked by their loss of power since Saddam Hussein's secular but Sunni-dominated regime was toppled.

US Secretary of State John Kerry said Washington was "very, very concerned" by the fighting, but would not send in troops.

But Iran did offer help to its neighbour. Deputy chief-of-staff General Mohammad Hejazi said Tehran would send military equipment and advisers if Baghdad requested them.

Iranian involvement would be sure to stoke tensions, as Sunnis accuse Tehran of backing the government's discriminatory policies.

But some Sunni militias have supported the government, with one group killing six militants in a firefight outside Fallujah.

Two soldiers were also killed in a militant attack on an army checkpoint in western Baghdad.

Tensions in Anbar have run high since December 28, when security forces arrested a Sunni MP on terrorism charges.

The government dismantled a months-old anti-government Sunni protest camp two days later, sparking deadly clashes.

Isil is also one of the strongest rebel groups fighting to overthrow Syrian President Bashar al-Assad, whose base is in the Alawite offshoot of Shi'ite Islam.