American forces are massing in large numbers around Baghdad following their probing raid into the city on Saturday.
The Americans lost at least one tank in Saturday's raid
A US armoured column of 2,000 vehicles crossed the River Euphrates towards the Iraqi capital at dawn on Sunday
The BBC's Peter Grant, who witnessed the advance of the column, says this force will join the two other brigade combat teams which are already in position near the city.
One, made up of the US 3rd Infantry, is based at Baghdad airport to the west of the city. Another, a US marine force, is in position on the south-eastern approaches to the city.
Units from the 3rd Infantry took part in the raid through the south-west of Baghdad on Saturday, driving up the main approach road into the city.
The column of 30 Abrams tanks and Bradley armoured fighting vehicles came under fire from Iraqi troops and militia fighters but made its superior firepower tell, destroying numerous Iraqi vehicles.
The Pentagon says 2,000-3,000 Iraqis were killed, while American losses were relatively light - one US soldier died and a handful were injured.
Information Minister Mohammed Saeed al-Sahaf, by contrast, said Iraqi forces had repulsed an American attack from the south, claiming: "We were able to chop off their rotten heads."
US Central Command, based in the Gulf state of Qatar, says a new air war strategy is being implemented over Baghdad, to provide direct cover for more incursions by US ground troops into the Iraqi capital.
They said Saturday's raid was intended to test Iraqi responses, while future operations will be aimed at specific targets.
Thousands of people are reported to be leaving Baghdad in advance of an expected American advance into the city.
The BBC's Rageh Omaar reports that there is nervousness in the city following Saturday's incursion by American forces.
He says there is very little traffic on the streets, the vast majority of shops are closed and there are very few civilians outside.
It was a relatively quiet night in Baghdad, which has been under aerial bombardment for more than two weeks, but by the morning there was the sound of artillery barrages from the southern outskirts of the city.
Iraqi television has reported that people will be banned from entering the city between 6pm and 6am local time from Sunday.
The US military said this would have "no impact" on their operations.
Much of the Iraqi capital is without electricity.
Increased numbers of Iraqi troops have been seen on the streets of the city.
Militia fighters dressed in black are on patrol and manning machine-gun positions at some of the main road junctions.