The military said that during the operations, the soldiers came under heavy small-arms fire and rocket-propelled grenade attacks in both places.
Jameel Mohammed, an Amara health department director, confirmed receiving at least 16 bodies and another 37 wounded people.
Major David Gell, a British military spokesman, said: "Iraqi special operations forces backed by multinational forces were involved in an operation in the Maysan province, but we are still awaiting details.
"Iraqi special operations forces led the mission in Amara, with multinational forces in support.
"The operation was carried out with the intention of arresting rogue militias who have been involved in assassinations and roadside bombings," he said.
He also said the militias had been involved in extortion and weapons trafficking.
Latif al-Tamimi, chief of the security committee on the Maysan council, called the operation a "catastrophe", accusing troops of firing randomly.
Gell told Al Jazeera: "It must be made clear; the operation was conducted under the directive of [Nuri] al-Maliki and the government of Iraq. Iraqi Special Operations Forces were very much in the lead."
|"It must be made clear; the operation was conducted under the directive of al-Maliki and the government of Iraq. Iraqi special operations forces were very much in the lead."|
Major David Gell, British military spokesman
The fighters are believed to be Shia fighters loyal to Muqtada al-Sadr, a populist Shia leader and prominent figure in Maysan province.
Auda al-Baharani, an al-Sadr spokesman in Amara, said: "The British forces carried out a mission using helicopters and aircraft in Amara. They used bombs, missiles and small arms fire."
An Iraqi security official said the British troops threw leaflets from helicopters declaring: "The Iraqi government will not be soft on terrorism" and "Maysan will not be a safe area for the Iranian Qods Force and its agents who want to weaken the Iraqi government".
The leaflets also carried pictures of fighters wanted by the security forces, he added.
Elsewhere on Monday, two people were killed in clashes between Iraqi police and al-Mahdi Army in Nasiriya, about 120km south of Amara, police said.
The fighting began after a number of police patrols were attacked on Sunday night, a police officer and an official in the town's health department said.
British forces handed security control of Maysan province back to Iraqi security forces in April.