An official in the office of Iraq's national security adviser confirmed the operations, which were also launched in the Tharthar area near Fallujah and in Diyala province.
Iraqi security forces were in the lead, backed by U.S. forces, the official said on condition of anonymity for security reasons. He refused to estimate the number of troops involved, or how long the operations would last.
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The U.S. military confirmed operations were under way but did not give more details.
"We started operations a couple days ago in some areas and we started operations this week in other areas," military spokesman Lt. Col. Christopher Garver said. He declined to be more specific, saying only that the operations were taking place across Baghdad and across Iraq.
Meanwhile, Iraqi and British forces fought a fierce battle with Shiite militiamen while conducting house-to-house searches early Monday in and around Amarah, 200 miles southeast of Baghdad. It was unclear whether the fighting was part of the new military operations.
A doctor at Amarah's general hospital said 36 bodies had been taken to his facility, though he could not determine how many were militiamen and how many were civilians. He spoke on condition of anonymity because he was not authorized to talk to media.
More than 100 people were injured in the fighting, and at least three of those killed were Iraqi policemen, police and hospital officials said.
The British military in Iraq could not immediately comment on the reports, but a Ministry of Defense spokeswoman in London said details of the fighting were still "quite sketchy" but that there were no British casualties.
The spokeswoman, speaking on condition of anonymity in line with ministry policy, said that the British soldiers played a supporting role to Iraqi security forces during the raid and fighting.
The U.S. military released a statement saying at least 20 insurgents had been killed and six wounded in coalition operations targeting "secret cells" in Amarah. Another suspect was detained, it said.
The men were believed to be members of a terror network that imports deadly armor-piercing weapons made in Iran known as "explosively formed penetrators," or EFPs, the statement said. They also were suspected of bringing militants from Iraq to Iran for terror training, it added.
Coalition forces came under small arms fire and rocket-propelled grenade attacks during the raids, and called in air support, the military said. The suspects were killed by fire from aircraft, it said.
The U.S. statement did not specify whether the coalition troops were American or British.
Iraqi police said the Mahdi Army, the militia commanded by radical Shiite cleric Muqtada al-Sadr, was involved in the clashes, which lasted for about two hours before dawn.
Amarah is the provincial capital of Maysan province, a predominantly Shiite region that borders Iran. Iraqi forces took over control of security from British troops there in April.
The city has seen intense militia fighting, most recently in October 2006, when the Mahdi Army briefly took control of the city and fought prolonged gunbattles with local police. At the time, Amarah's police force was believed to be dominated by a rival militia, the Badr Brigades. More than 30 people were killed in the standoff.
Elsewhere Monday, eight people were killed in clashes that erupted between Iraqi police and Mahdi Army fighters in Nasiriyah, about 120 kilometers (70 miles) south of Amarah, police said. More than 60 people were injured, most of them police, they said.
The fighting began after some police patrols were attacked there Sunday night, a police officer and an official in the town's health department said, both on condition of anonymity out of security concerns.
Some local tribesman had joined the fight, siding with Iraqi police in trying to oust the militiamen from their town, the officials said.
The battle included at least eleven mortar strikes on police headquarters in Nasiriyah, the officials said.
Clashes continued through Monday, and local authorities imposed an indefinite curfew on the city, police said. By early afternoon, the fighting had spilled over into the Souk al-Sheikh area south of Nasiriyah, and into al-Rifaie, north of the city, police said.
In Baghdad, two parked car bombs exploded near a gas station in southern Baghdad, killing at least seven people who had been lining up to buy fuel, police said. Up to 25 others were injured, and four cars were incinerated by the blasts, they said.
Nearby, gunmen ambushed an Interior Ministry convoy, killing an Iraqi colonel and his two guards, police said.
Also Monday, four civilians were killed and 13 injured when a parked car bomb ripped through a busy vegetable market in Fallujah, 65 kilometers (40 miles) west of Baghdad, police said.