Iraq's prime minister has urged the city of Fallujah to expel al-Qaida militants to avoid an all-out battle - possibly signalling an imminent military move to retake the former insurgent stronghold.
Iraqi government troops have surrounded the city, which lies in the western Sunni-dominated Anbar province and which was overrun by al-Qaida fighters last week.
Mr Al-Maliki did not say how he expects Fallujah residents and pro-government tribesmen to push the militants out. In his message, broadcast over state TV, he also urged Iraqi troops to avoid targeting Fallujah's residential areas.
Along with Fallujah, al-Qaida fighters last week also took control of most parts of the provincial capital of Ramadi.
Iraqi troops have since been trying to dislodge militants from the group, known as the Islamic State in Iraq and the Levant, or ISIL, from the two cities. On Sunday, fighting in Anbar killed at least 34 people, including 22 soldiers.
The recent gains by al-Qaida in Iraq have been a blow to the country's Shiite-led government, as sectarian violence has escalated since the US withdrawal. US Secretary of State John Kerry said Washington was "very, very concerned" by the fighting but would not send in American troops.
The Iranian army's deputy chief-of-staff, Gen. Mohammad Hejazi, said Iran was also ready to help Iraq with military equipment and advisers, should Baghdad ask for it. Any Iranian help would exacerbate tensions as Iraqi Sunnis accuse Tehran of backing what they say are their Shiite-led government's unfair policies against them.
Al-Qaida fighters and their supporters are still controlling the centre of the city where they can be seen on the streets and around government buildings. Al-Qaida black flags have been seen on government and police vehicles captured by the militants during the clashes.
In Ramadi, sporadic clashes were taking place in some parts of in and outside the city on Monday, residents there said.
Dozens of families were fleeing the two cities to nearby towns, crammed in cars loaded with their belongings.
On Sunday at least 22 soldiers and 12 civilians were killed, along with an unknown number of militants, and 58 people were wounded during clashes between al-Qaida fighters on one side and the army and its allied tribesmen on the other.
Tensions in Anbar have run high since December 28, when Iraqi security forces arrested a Sunni MP sought for terrorism charges. Two days later, the government dismantled a months-old, anti-government Sunni protest camp, sparking clashes with militants.
ISIL is also one of the strongest rebel units in neighbouring Syria, where it has imposed a strict version of Islamic law in territories it holds in the civil war raging there. It also has kidnapped and killed dozens of people it deems critical of its rule. On Saturday, it claimed responsibility for a suicide car bombing in a Shiite-dominated neighbourhood in Lebanon.
Iraq's al-Qaida branch has fed on Sunni discontent and on Syria's civil war, in which mostly Sunni rebels fight the government of President Bashar Assad whose base is a Shiite offshoot sect.