Local police and their Sunni tribesmen allies say they defeated Al-Qaeda-linked militants on January 10 in the southern Malaab and Fursan areas of Ramadi, killing some 25 fighters in battles.
Twenty-two insurgents were killed in fighting east of Ramadi, with at least four members of an Iraqi SWAT team being killed during clashes.
Battles were also fought in Albu Bali, a village between Ramadi and Fallujah, according to an RFE/RL correspondent in Ramadi.
Abdulkhalqi Muhammad earlier reported that there was tank shelling, aerial bombardments, and car-bomb explosions in Ramadi overnight.
Security forces, allied with fighters from local Sunni tribes, have been trying to expel antigovernment Sunni insurgents and Al-Qaeda-linked fighters who overran parts of Ramadi and the nearby city of Fallujah last week.
The Iraqi Army has held off on launching a major offensive to retake the two cities, fearing civilian casualties could incite Sunnis to side with the Al-Qaeda insurgents.
But Defense Ministry spokesman Muhammad al-Askari told AP on January 10 that "if there is no other solution, then the security forces and allied tribal fighters will enter these cities."
Ramadi 'Back To Normal'
In Ramadi, several tribal sheikhs and local government officials were targeted with car bombs. Casualty tolls from those attacks and the latest violence aren't available.
Most of Ramadi is reportedly returning to normal, although the Al-Qaeda insurgents are still holed up in some southern neighborhoods in the city, where some Al-Qaeda black flags are visible, RFE/RL's correspondent reports.
But he says clashes in Ramadi are sporadic and that Iraqi security forces have bombarded the militants with artillery and mortar fire.
The Al-Qassim Bridge, which leads to Ramadi's city center, has been blown up in the fighting.
But the situation in Fallujah is far less stable, with armed clashes occurring frequently and people fleeing the city.
Fallujah resident Mundhir Attaullah described the situation as he left the city with his family.
"The army is shelling residential areas of the city. When the shells land, they land randomly and there are areas where there aren't any militants present, so they land on families who have caused no trouble," Attaullah said.
"The people cannot stay in their houses and therefore we decided to flee. The road to Baghdad is closed and the areas around Ramadi are out of bounds, we had no choice except this route, so we came here."
There are also shortages of electricity, water, gas, and foodstuffs in many parts of the city, according to another Fallujah resident, and "the nearby medical clinic was closed, as there was no one to work."
But some parts of Fallujah, which has a population of some 325,000, are returning to normal, another resident said.
"The situation is good and the market is stable. All of the shops are open and we ask all the families who left to return to Fallujah," he said.
Clashes have also occurred the Gurma administrative district, about 30-35 kilometers from Fallujah, as well as in Saqlawia, which is linked administratively with Fallujah.
At the UN, the Security Council gave its backing to the Baghdad government in its attack to regain control in Ramadi and Fallujah and condemned the militants for launching their raids in Anbar last week.