US-led forces claim to have secured almost three quarters of Falluja, but rebels struck back, kidnapping three relatives of the prime minister and taking 20 Iraqi soldiers hostage.
Militants threatened to behead interim Prime Minister Iyad Allawi's relatives if he refused to call off US-led "Operation Fair Dawn" to wrest control of Falluja from insurgents.
The head of Iraqi forces in Falluja said they had found "slaughterhouses" where hostages had been held and killed, along with records of victims and hundreds of CDs.
But Major-General Abdul-Qader Jassim told reporters he could not say if there was any clue of the fate of at least nine foreign hostages still missing.
Rebels, in a video given to news agencies, said they had snatched more than 20 Iraqi National Guardsmen in Falluja, but did not say how and did not make any specific threats.
The footage showed masked guerrillas pointing assault rifles and rocket-propelled grenade launchers at a group of men with their backs turned dressed in National Guard uniforms.
Air strikes, shelling and mortars shook parts of Falluja during intense clashes interspersed with periods of relative calm, a reporter in the Sunni Muslim city said.
US forces said they bombed a mosque after being fired on.
"The fighting, as you all know, in an urban area is very close and very violent," the US Marine commander at Falluja, Lieutenant-General John Sattler, told reporters, adding the insurgents could no longer coordinate their resistance.
"They are now in small pockets, blind, moving throughout the city. We will continue to hunt them down and destroy them."
Senior US commanders said today most rebel leaders had slipped out of the city, leaving the command of the remaining 2,000-3,000 fighters to relatively junior ranks.
But Mr Allawi and US commanders claim it is vital to recapture Falluja from foreign militants led by al Qaeda ally Abu Musab al-Zarqawi as well as Saddam Hussein loyalists so national elections can go ahead in January.
Mr Allawi's 75-year-old cousin, Ghazi Allawi, his cousin's wife and their daughter-in-law were seized near their home in Baghdad today, Mr Allawi's spokesman said.
The previously unknown Ansar al-Jihad group said the hostages would die unless Mr Allawi, "head of the Iraqi agents", halted the offensive that began on Monday and freed prisoners.
"If the agent government does not meet our demands within 48 hours we will behead them," it said in a statement dated today and posted on an Islamist Web site.