Iraq's Prime Minister Nouri al-Maliki has urged all those who joined al-Qaeda and now face a siege by his troops in the western Anbar province to give up their struggle, hinting of a possible pardon if the militants give up the fight.
Fallujah and parts of the Anbar provincial capital Ramadi farther west have been outside government control for days - the first time fighters have exercised such open control in major cities since the height of the insurgency that followed the 2003 US-led invasion.
Speaking in a televised address on Wednesday, Maliki vowed to continue the "sacred war'' against al-Qaeda-linked group the Islamic State in Iraq and the Levant (ISIL) and finish the push to retake the Anbar cities.
He said he is calling on those who were “lured to be part of the terrorism machine led by al-Qaeda to return to reason".
If they do so, Maliki says his government will "open a new page to settle their cases."
On Wednesday, fighting broke out two areas of Fallujah - Al-Askari in the east and Al-Shuhada in the west - and lasted for about an hour, witnesses said. Both neighbourhoods were also shelled, they added.
Later in the day, lightly armed uniformed traffic police returned to the streets of Fallujah to direct vehicles and control intersections, but gunmen remained in control of the city.
ISIL has been active in the city but so have anti-government tribes.
The security forces have, meanwhile, recruited their own tribal allies in the fighting that has raged in Anbar province for more than a week and killed more than 250 people.
The renewed unrest in Anbar was sparked last month by the arrest of a Sunni politician sought on terrorism charges, followed by the government's dismantling of an anti-government Sunni protest camp in Ramadi.
On Tuesday, a spokesperson for the ISIL urged Sunni fighters to keep up their fight against the Shia-led government of Maliki.
"Oh Sunni people, you were forced to take up the weapon," Abu Mohammed al-Adnani said in an audio recording.
"Do not lay the weapon down, because if you put it down this time, the (Shia) will enslave you and you will not rise again.”
The defence ministry on Tuesday said that the military launched missile strikes on Ramadi and killed 25 fighters.