"I think the sovereignty will be weak and not complete," said Mahmoud Othman, a member of the Governing Council. For "the security situation, there will still be the United States," he said.

He also expressed worries there will be limits as to what laws the Iraqis can pass. If the government can't make laws or provide security "it will not be real sovereignty," he said. "The less sovereignty there is, the less the possibility that the government will be able to work and achieve its tasks."

U.N. envoy Lakhdar Brahimi has proposed that the Governing Council be dissolved and caretaker government made up of nonpartisan experts be created to run Iraq until elections in January. Washington has said that since Iraqi security forces are still not able to fight insurgents, U.S. forces will hold security powers even after the handover.

Ahmad Chalabi, a council member and close American ally, said he is demanding from top U.S. administrator L. Paul Bremer that the coming government be given strong roles on both the security and political fronts.

"We tell him that Iraqis should have a bigger role in security, we tell him that Iraqis should have a bigger role in taking financial decisions, we tell him that Iraqis should have a role in running the Iraqi reconstruction fund," he told the Arab television station Al-Arabiya

John Negroponte, nominated to be ambassador to Baghdad, said at his Senate Foreign Relations Committee hearing that Iraqis will have "a lot more sovereignty than they have right now" after the June 30 handover, but the United States will still have a key role in providing and overseeing security, and the caretaker government won't be able to make laws.

Negroponte said the focus of the transitional government would be to organize elections, and the cabinet ministries will carry out the government's day-to-day operations.

White House spokesman Scott McClellan said there is "ample precedent for self-imposed limits on authority of interim, caretaker governments, such as likely to be the case here in this first phase of Iraq's transition to democracy."

Also Tuesday, a Red Cross team visited Saddam Hussein to see his conditions in U.S. custody, Kimmitt said, but he refused to say where the visit took place. It was the first since the Red Cross visited the ousted Iraqi leader in February.

The battles in the south Monday evening took place on the east side of the Euphrates River, across from Kufa and Najaf, Kimmitt said.