Mr Sadr is surrounded by hundreds of his heavily armed militia, known as the Mehdi Army, at his office near the Imam Ali shrine in the heart of the holy city of Najaf. The US-led coalition has branded him an "extremist" and is seeking his arrest.
Brigadier General Mark Kimmitt, deputy director of operations for the US occupation forces in Iraq, said on Wednesday that the US was thinking long and hard about how to bring the situation in Najaf under control, but did not wish to alienate pilgrims visiting the holy city.
But he vowed that "offensive operations" to destroy the Mehdi army would continue. They will be "deliberate, precise and powerful, and they will succeed", he said.
Coalition forces fought an urban guerrilla war on two fronts on Tuesday, bombarding the Sunni town of Falluja while clashing with pro-Sadr militiamen whose support they fear could spiral if the rioting is not quickly brought under control.
About 30 US troops and more than 100 Iraqis have been killed throughout Iraq since the rioting began over the weekend.
Responding to the violence, US President George W. Bush and Tony Blair, UK prime minister, said their commitment to a peaceful and democratic Iraq was "unshakeable".
"Our resolve is firm... and we will prevail,'' Mr Bush was reported to have said through his official spokesman after being briefed about developments at his ranch in Crawford, Texas.
A US military spokesman said 1,200 American and Iraqi forces had advanced to the centre of Falluja on Tuesday, where four US contractors were savagely killed and mutilated last week, before retreating to the outskirts under fire. Helicopter gunships fired back.
Sunni insurgents appeared cheered that Shia Iraqis could finally be joining in their revolt. The fighting spread to Ramadi, where US marines exchanged fire and killed three Iraqis, local journalists said.
A US defence official said that up to 12 US marines had been killed near the governor's palace in Ramadi in Anbar province on Tuesday night.
Three more US soldiers were killed in attacks near Baghdad's Shia shrine of Kathimiya, a spokesmen said.
British and Italian forces fought street battles in the southern towns of Amarah and Nasiriya respectively. Reports from Nasiriya said 20 Iraqis were killed.
For the first time since the end of the war, widespread Shia violence has focused attention on the non-US coalition nations that have sent troops to Iraq.
These contingents are based almost entirely in the Shia-dominated south. A Salvadorean and a Ukrainian were among those killed in the fighting since Sunday.
Twelve Italian soldiers were also wounded on Tuesday as they tried to recapture bridges over the Euphrates from Sadr loyalists, according to Italian military officials. Militiamen said they also took two South Korean human rights activists hostage, but later released them.
British officials said they had fought 18 skirmishes in a second day of clashes in Amarah near the Iranian border. Twelve Iraqis have been killed in two days of fighting.
But in Basra, where Mr Sadr commands less support, British mediators negotiated a withdrawal of militia who had set up rocket launchers on the roof of the governor's office.
The Spanish defence ministry said its forces had clashed with Mr Sadr's in Diwaniyah town, east of Najaf.
Hospital officials in the Baghdad suburb of Sadr City said the numbers of Iraqi dead in the capital in two days of clashes had risen to at least 57, and 236 wounded.
US officials said that operations to crush the dual insurgency would continue, as some Governing Council members appealed for a negotiated settlement.
"There is no question we have control of the country," said Paul Bremer, US governor of Iraq, in an interview with CNN.
Adam Ereli, US State department spokesman, said: "What we're witnessing, frankly, is efforts by a small and vocal minority to resist the rule of law. I don't think we're seeing anything that's either widespread or shared by a large part of the population. I would say there's no reason, no cause to shift the deadline based on what's going on." Advisers to members of the Governing Council said Mr Bremer had ruled out a negotiated settlement with Mr Sadr.
"Mr Bremer will not accept less than the arrest of Moqtada Sadr," said Adnan Ali of the politiburo of the Dawa party, which had negotiated with Mr Sadr earlier in the day.
The Governing Council, which is desperate to restore order lest the June 30 handover be postponed, cancelled a five-man delegation they had been preparing to send to Najaf. They cited security concerns, but other reports said Mr Sadr had rebuffed the delegation.