The Jem rebels are thought to have crossed into Sudan from bases in Chad
Sudanese troops have clashed with rebels from Darfur outside the capital, Khartoum, reports say, prompting an overnight curfew in the city.
The rebels announced that their forces had taken nearby Omdurman and were now fighting in the city, but the government said they had been defeated.
Sudan's army had been preparing for a possible attack on the capital after detecting a heavily-armed rebel column.
Correspondents say this is the closest the rebels have ever been to Khartoum.
A curfew was imposed in Khartoum from 1700 local time (1400 GMT) until 1000 on Sunday, and roads and bridges over the Nile are reported to be closed.
The US National Security Council spokesman Gordon Johndroe said the White House was very concerned about the fighting.
"We would urge that both sides cease hostilities, whether it is the rebel group or any response from the government. We want to see a calm and order restored."
The rebels, from the Justice and Equality Movement (Jem) claimed to have taken the Wadi Saidna air force base about 10 miles north of Khartoum, Omdurman and to have entered the capital.
But Sudan's presidential adviser Gazi Salaheddin told the BBC the rebels had now been defeated except for a few groups.
Mr Salahddein said Jem had tried to take the capital with the backing of Chad's government. He warned that another contingent of rebel fighters might be on their way across the border.
A Jem spokesman denied these reports and said they were still on track with their plan to overthrow the government.
The BBC's Amber Henshaw, in Khartoum, says the atmosphere in the city is tense but quieter than during the height of the fighting when artillery fire could be heard.
Local people in the area said the fighting had lasted several hours. Some reported power failures and street fighting.
A BBC News website reader in Omdurman said the rebels had cut off access to food supplies.
"The fighting here is terrible," said Chingnachcook. "They are everywhere. I have smuggled out my laptop and I have managed to hide up in our loft."
Asabusaid, in Khartoum said: "Fighter aircraft have been observed taking off from Khartoum and the sound of attack helicopters can be heard in the distance from time to time."
Hashimi, also in Khartoum said: "There are several army men on each street and there is lots of fighting. No one is sticking to the curfew and many are at the mosques."
Experts say Chad and Sudan are fighting a proxy war using each others rebels to achieve their military objectives.
The Jem is one of several rebel groups fighting the government and pro-government Janjaweed militia in the western Darfur region since 2003 over alleged discrimination by the authorities in favour of Arabs.
The rebels have been involved in raids on government forces in the area before.