Iraq's prime minister has urged residents and tribes in Anbar province to "expel" al-Qaeda-linked fighters to avoid an all-out battle - remarks that may signal an imminent military move to retake the former insurgent stronghold.
In a message broadcast over state TV on Monday, Nouri al-Maliki told people to push fighters of the Islamic State of Iraq and the Levant (ISIL) out of Fallujah, and urged the Iraqi army to avoid targeting the city's residential areas in the fighting.
Tension remains high in Anbar province amid reports of sporadic clashes taking place in some parts in and outside the city of Ramadi, and the killing of a number of al-Qaeda-linked fighters in a firefight with a pro-government group outside Fallujah.
Fallujah residents said clashes continued into early morning on Monday along the main highway that links the capital, Baghdad, to neighbouring Syria and Jordan.
Fighters from ISIL and their supporters are still controlling the centre of the city where they can be seen on the streets and around government buildings.
Al-Qaeda black flags have been seen on government and police vehicles captured by the fighters.
The army has besieged Ramadi, the capital of Anbar province, and has launched air strikes on the city.
Ahmed Abu Risha, a senior Sunni tribal leader and head of the Awakening Council in Anbar province, told Al Jazeera that an agreement has been reached between tribal groups and the Iraqi government to withdraw the army from Ramadi and Fallujah.
"The ISIL rebels were trying to bring the battle to the cities of Anbar and Fallujah because there's no security there and they know they will be thoroughly defeated in the desert," Abu Risha said.
Al Jazeera's Iraq correspondent Imran Khan said: "Sheikh Abu Ahmed Risha is going to lead his fighters inside [Fallujah] with backup from the Iraqi army. There are rival tribes inside Fallujah who are sympathetic to ISIL. Not only will he have to fight with the ISIL fighters, but he'll also have to take on rival tribes. It's going to be a very tough fight for both him and the Iraqi army."
The takeover of Fallujah and parts of Ramadi is the first time that tribal fighters have exercised such open control in major Iraqi cities since the height of the bloody insurgency that followed the US-led invasion of 2003.
Dozens of families fled the violence on Monday, heading towards the city of Kerbala, and have sought refuge in schools in neighbouring towns.
Mohammed al-Khuzayee, deputy general secretary of the Iraqi Red Crescent, said three truckloads of food and sanitary items were being shipped to them.