In the capital Baghdad, to the south of Tikrit, a US military commander said six US soldiers believed to have been prisoners of war had been rescued from Iraqi forces.
In the streets of Tikrit - the last major Iraqi city not controlled by US forces - no regular Iraqi soldiers were seen, but tensions were running high as residents toted Kalashnikov assault rifles and grenades.
The armed men said they were ready to surrender to advancing US forces, but only if Iraqi opponents of Saddam's regime - notably Kurds and Shiites - did not accompany them into the city, located 180 kilometres (110 miles) north of Baghdad.
Saying that much of the city's population of 100,000 had fled, the residents said they were carrying weapons to protect themselves from looters.
Fifteen tribal leaders called for an end to US bombardment of the city, the Iraqi leader's hometown, so that the peaceful surrender of pro-Saddam militia there could be negotiated, one of them said.
Earlier on Sunday, a team of journalists from the CNN news network twice came under fire from unidentified gunmen, once as they passed through a checkpoint and a second time as they raced out of town, in a dramatic scene seen live.
The fire fight seemed to indicate that at least part of Tikrit was still under the control of Saddam's regime, on day 25 of the US-led war in Iraq to oust him and strip Iraq of its alleged weapons of mass destruction.
Once a dusty farming town, Tikrit was transformed into a gleaming modern city with ornate palaces and mosques, and opulent villas for Saddam's trusted aides, after his Baath party took power in 1968.
The fall of the city to US-led forces would mark a significant step towards the end of the conflict in Iraq.
In good news for the US forces, military officials said six US soldiers had been rescued from Iraqi forces north of Baghdad.
"They are being ferried now to a shock trauma platoon in Baghdad urgently for attention," said a commander in Baghdad asking not to be named.
The rescue of the six unidentified soldiers was confirmed by US General Tommy Franks who told CNN they were in good shape.
But lawlessness and looting continued to plague much of the country, stoking a sense in insecurity in Baghdad, the northern oil-rich towns of Kirkuk and Mosul, and Basra in the south.
US forces, facing mounting anger among Baghdadis for failing to stem days of looting since taking the capital on Wednesday, set up an operations centre in the city centre to recruit Iraqi workers for key sectors.
"We want workers, not only senior officials," said Gunnery Sergeant Claudia Lamantia, of the First Marines Expeditionary Force. "The idea obviously is to get everything back running."
Baghdad, which has five million residents, has been without electricity for about 10 days, and most homes are also without running water and telephone services. Public transportation is non-existent.
More people were out in the streets of Baghdad on Sunday, but fears over security persisted, after the ransacking in recent days of entire sections of the capital by rampaging youths from the Shiite suburb of Saddam City.
Volunteers were also out in force to remove the corpses of those killed in fighting between US troops and Saddam loyalists, AFP reporters said.
In an ominous sign of possible violence still to come, US military officials said Sunday that Marines had uncovered 310 vests fitted for use by suicide bombers, with about half of them "engineered with explosives".
US military commanders meanwhile set their sights on Tikrit and the surrounding area, with Captain Frank Thorp telling reporters at the US Central Command's forward operating base that US Marines were operating in the area.
"Task Force Tripoli has moved north and is currently conducting operations in the vicinity of Tikrit," Thorp said.
Coalition war planes have been pounding Iraqi positions in the Tikrit area for more than a week, trying to wear down the remnants of Iraq's elite Republican Guard, the country's most formidable and most loyal forces.
In northern Iraq, a move to co-opt the existing police force in the oil-rich city of Mosul sparked an angry reaction from Kurdish residents, who had greeted the collapse of Saddam's government with jubilation.
Since Kurdish rebel fighters entered Mosul on Friday, the city of 1.5 million people has been rocked by ethnic violence between the Kurds and Arab residents which hospital officials say has killed as many as 20 people.
In the northern oil capital of Kirkuk, US troops were deployed outside the governor's office, in a sign they were steadily taking over control of the city from Kurdish forces as demanded by neighbouring Turkey.
In central Iraq, gunmen in the holy city of Najaf surrounded the house of Ayatollah Ali Sistani, Iraq's leading Shiite cleric, ordering him to leave the country within 48 hours, a cleric in Kuwait said.
Both Britain and the United States announced they were beginning to scale back their military presence in the Gulf now that the strike phase of the war was over.
But anti-war demonstrations continued around the world on Sunday, with 10,000 protestors marching through central Sydney calling for peace and the immediate withdrawal of Australian troops from Iraq.
Elsewhere, US Secretary of State Colin Powell warned Syria against offering safe haven to members of Saddam's ousted regime.
On Saturday, Saddam's top weapons advisor General Amer al-Saadi surrendered to US troops in Baghdad, saying he was happy to be questioned because Iraq had none of the banned weapons the United States used to justify the war.