The army said in a statement read out on state television that its troops were under rebel attack in Omdurman, the capital's twin city to the northwest joined to Khartoum by a bridge, where witnesses reported heavy fighting.
"Army forces are currently facing attack by rebels loyal to (JEM leader) Khalil Ibrahim in the north of Omdurman," the military said.
The rebel Justice and Equality Movement (JEM) said its fighters had taken control of Wadi Saidna air force base about 10 miles (16 kilometres) north of Khartoum, but this was impossible to verify.
"According to the information I have JEM forces are in the capital right now. They took Wadi Saidna air force base and they moved down into the capital," London-based JEM spokesman Ahmed Hussein Adam told AFP by telephone.
Witnesses said there was heavy fighting in Omdurman for at least two hours.
Omdurman resident Sadiq Babo Nimr told AFP by telephone that frightened residents were cowering at home amid heavy artillery fire.
"It's just outside my flat, I can hear bombardment from very heavy artillery," he said.
"We're just lying down, there are stray bullets around the building, my wife and children are very scared."
He said that he heard distant gunfire and the fighting drew closer at around 4:30 pm (1330 GMT). He said there was no electricity and mobile phone networks were down.
Heavily armed troops patrolled the capital's deserted streets after shops shut, another witness told AFP by telephone.
A curfew was imposed in the capital from 1400 GMT on Saturday until 0300 GMT on Sunday, the interior ministry said, urging Khartoum residents to say indoors and be vigilant.
It was not immediately known if there were any casualties in the fighting.
The London-based spokesman said JEM wished to topple the regime in Khartoum.
"We don't want this suicidal regime to remain in power and to kill and massacre our people every day," Adam said.
"We reiterate our position to cooperate with the entire international community in democratic transformation and find a comprehensive and genuine solution to the conflict in Darfur," he added.
State media said the Sudanese army had repulsed an attack by Chadian forces on the border in the Kishkish area, with an army spokesman saying the Chad raid was a bid to create a diversion from JEM's advance on Khartoum.
"Our armed forces stood up to the Chadian forces, inflicted casualties on them and drove them back into the Chadian territories," Brigadier Mohammed Osman al-Aghbash told Omdurman Radio.
He accused the Chadian government of providing "direct support" to JEM's "subversive operation" in Khartoum.
A Western diplomat who requested anonymity said that Sudan had on Friday warned embassies in the capital of the rebel advance.
"There are rebels that have been moving eastwards from the Chadian border," one diplomat quoted the authorities as saying.
Extra security forces began deploying on Khartoum's streets on Friday, the Sudanese authorities said.
JEM are "trying to prove a point. I don't think anyone expects at this stage to see heavy fighting in the streets. I think they're going to be crushed but the simple fact that they got to Omdurman is huge," the diplomat said.
Sudan and Chad accuse each other of backing rebels battling to topple their respective regimes.
JEM has carried out a number of high-profile attacks in Sudan, including raids on Chinese-run oil fields, criticising what it calls Beijing's refusal to rein in Khartoum's alleged human rights abuses.
JEM was involved for years with another rebellion in eastern Sudan by the disenfranchised Beja and Rashaida people, although the Eastern Front rebel alliance signed a peace deal with Khartoum in 2006.
The group has emerged militarily the most powerful rebel movement in Darfur, where fighting groups have fractured into myriad organisations during the past five years, and JEM has always promoted national objectives.
More than two million people have fled their homes in the western Sudanese region of Darfur since the Khartoum government enlisted militia allies to put down a revolt in the region in 2003.
UN humanitarian chief John Holmes said last month that the death toll in Darfur from five years of war, famine and disease might have reached 300,000. Khartoum says the toll is around 9,000.