The US has warned that the new government due to take power in Iraq on 1 July will have to delegate some of its powers to the coalition.
Critics of the handover plan say it is not offering full sovereignty
Secretary of State Colin Powell said it was important that coalition troops continued to operate under US command.
His comments came after a day of heavy fighting in Falluja, where US soldiers and Iraqi security forces had been due to begin joint patrols on Tuesday.
The BBC's Jennifer Glasse in Fallujah says the patrols have been delayed.
The patrols are now expected to begin later this week.
At least one US soldier and eight Iraqi fighters were killed in Monday's violence, the US said.
US officials said marines retaliated when militiamen opened fire with rockets, mortars and machine-guns from a mosque.
New unrest was also reported in Kufa, outside the Shia holy city of Najaf, and an explosion in Baghdad killed two US soldiers.
The US has insisted it is sticking to the planned timetable of transferring power to an interim Iraqi government after 30 June despite the upsurge of violence in Iraq.
But in an interview with Reuters news agency, Mr Powell said that while the new government would take full sovereignty over the country, it would have to give some of it back to the Americans so that the US would still be in command of its own troops.
"I hope they will understand that in order for this government to get up and running - to be effective - some of its sovereignty will have to be given back, if I can put it
that way, or limited by them," Mr Powell said.
"It's sovereignty but [some] of that sovereignty they are going to allow us to exercise on their behalf and with their permission."
'Army of occupation'
Speaking in Washington, Mr Powell said the coalition did not mean to "seize anything away" from the planned caretaker government.
"It is with the understanding that they need our help and for us to provide that help we have to be able to operate freely, which in some ways infringes on what some would call full sovereignty," he said.
But in an interview with the Associated Press, the current head of the US-backed Iraqi Governing Council, Massoud Barzani, accused the US of behaving like "an army of occupation".
"If it were me, I wouldn't have allowed it to come to this by making earlier
mistakes," Mr Barzani said.
"I probably wouldn't have made the mistake of letting an army of liberation turn into an army of occupation."
The veteran Kurdish leader, who holds the rotating presidency of the Iraqi Governing Council for April, added that the US-led invasion had helped the Kurds to "accomplish a major part" of their rights.
The BBC's State Department correspondent, Jon Leyne, says there is a danger that the new Iraqi government will be seen as a puppet of the American occupiers.
Mr Powell insisted that the coalition wanted the new government to have a real part to play in rebuilding Iraq:
"From day one on, I hope that they will exercise more and more control over the ministries, over the priorities for reconstruction, over answering questions of their people. It is in our interest to have that happen."
Earlier, Mr Powell said a compromise was being sought in Falluja despite the failure of a US truce which was extended for two days from Sunday.
Talks were under way with civic and tribal leaders in the surrounded city to arrange joint patrols with US and Iraqi forces, he said.
The aim, Mr Powell said, was to win the confidence of the citizens and to persuade gunmen to lay down their arms.