American-led forces in Falluja say that they are in control of almost all the city after six days of fighting.
US soldiers say house searches have yielded many weapons caches
They say just a few small groups are still firing in the south of the city, but reporters give differing accounts about the extent of their control.
However, the escalation of violence in other Iraqi cities has continued.
The first aid convoy arrived in Falluja amid fears of a humanitarian crisis. But US forces have prevented supplies being distributed, an aid agency said.
Fardous al-Ubaidi, from the Iraqi Red Crescent, said five lorries and three ambulances had driven to the city's hospital carrying food and medicine, but were told to go no further.
Aid agencies have expressed growing fears about the plight of civilians trapped in the city without food or medical supplies.
But a US commander reportedly said the US military was taking its own supplies into the city.
Marine Colonel Mike Shupp said casualties could be brought to the hospital, and cast doubt on the idea there were any civilians trapped inside the city, according to Reuters news agency.
In other developments:
- US and Iraqi security forces are battling to regain control of the northern city of Mosul, after insurgents launched attacks on two police stations there, reportedly killing at least six Iraqi National Guards
- US tanks and helicopters fire on suspected insurgents in the key oil city of Baiji, north of Baghdad
- Fire rages at an oil pipeline near the town of Taji, north of Baghdad, following an act of sabotage on Saturday night
- A senior official of the Iraqi Communist Party, Waddah Hassan Abdel Amir, is killed by gunmen in northern Iraq. The party is part of the interim Iraq government.
- The Los Angeles Times says one in six US soldiers returning home from Iraq is suffering psychological trauma.
Correspondents say the escalation of violence across Iraq demonstrates insurgents' determination to open up new fronts.
'Occupied, but not subdued'
One senior US officer said Falluja was "occupied, but not subdued".
Commanders say a house-to-house operation is sweeping from the north to the south of the city and could take another four or five days to complete.
In the last two days, up to 400 insurgents have been arrested, they say.
An AFP news agency journalist embedded with a unit of the marines - and speaking under military restrictions - told the BBC that American soldiers were searching every street corner, house and flat in their part of the city, though small pockets of resistance remained.
Patrick Baz said Falluja looked like a ghost town - with shop fronts riddled with bullets and tangled electrical wires dangling over deserted streets.
But Fadhil Badrani, a journalist operating independently in the city, told the BBC Arabic service that US-led forces had taken over only the northern districts.
He said in other parts of the city they controlled only the main roads.
He said heavy bombardment of the city, using aircraft, artillery and tanks is continuing.
Marines found what they thought was the mutilated body of a Western woman in the city.
Two Western women are known to be among those still missing in Iraq after being kidnapped.
The US marines say several large arms caches have also been found and destroyed.
Thirty-one US soldiers and six Iraqi soldiers have died in the battle for Falluja - including three US soldiers killed searching one house which was booby-trapped and blew up as they entered.
Earlier, commanders said that more than 1,200 insurgents had been killed since the assault began on Monday.
It is not known how many civilians have died. The Iraqi interim Prime Minister Iyad Allawi has insisted no civilians have been killed, but this seems unlikely given the intensity of the fighting, say correspondents.
Tens of thousands of residents fled before the offensive, but thousands of people are thought to have stayed.
The Iraqi Red Crescent reports that civilians are hiding in their houses, without drinking water and running low on food.
Mustupha Ahmed Abed, 2, lost his left leg in the fighting
"Our situation is very difficult," one resident in the centre of the city, Abu Mustafa, told Reuters news agency by telephone.
"We don't have food or water. My seven children all have severe diarrhoea. One of my sons was wounded by shrapnel last night and he's bleeding, but I can't do anything to help him."
The US-led assault on Falluja is aimed at stabilising the country before planned elections in January.