U.S. and Iraqi troops have pushed into central Fallujah, but a senior U.S. commander says there are days of tough urban fighting ahead to retake the city from insurgents.

Lieutenant-General Thomas Metz says coalition forces destroyed the insurgents' outer ring of resistance during the first two days of the assault. But, he warned of continued fighting. "The fight in Fallujah is far from over," he said.

"One of the key things about war is you never underestimate the enemy. There's no doubt that he is a smart enemy and has shown discipline for the most part," said one soldier.

Thousands of soldiers are now fighting small groups of insurgents in street by street battles.

"The enemy is concentrating on Iraqi security forces, knowing as they grow their capacity, that is the thing that they will have to fight in the future, and they are working hard to intimidate the Iraqi security forces," Lt. Gen. Metz said.

The U.S. military says 10 American troops and two Iraqi soldiers have been killed in the assault. U.S. commanders say the number of insurgent casualties is higher than expected, but they provided no numbers

Most of Fallujah's residents fled before the offensive. Lieutenant-General Metz says the most wanted man in Iraq, terrorist leader Abu Musab al-Zarqawi, likely fled the city before the assault.

Thair Al-Nakib, a spokesman for Iraqi interim Prime Minister Ayad Allawi's office, called for the insurgents to put down their weapons and save Fallujah from further military confrontations. "The Iraqi military troops await on standby to enter the city peacefully, and take control after the armed insurgents and terrorists put down their weapons," he said.

President Bush praised U.S. and Iraqi troops for bringing the insurgents to justice and advancing democracy.

The chief of the Association of Muslim Scholars, Harith al-Dhari, issued the call Tuesday -- one day after U.S. forces pushed into the mainly Sunni city.