The army rejected assertions by the Rapid Support Forces (RSF) that they had seized the presidential palace, the army chief's residence and airports in Khartoum and the northern city of Merowe.
Army chief General Abdel Fattah al-Burhan told Al Jazeera TV: "We think if they are wise they will turn back their troops that came into Khartoum. But if it continues we will have to deploy troops into Khartoum from other areas."
The RSF leader, General Mohamed Hamdan Dagalo, better known as Hemedti, called Burhan a "criminal" and a "liar."
"We know where you are hiding and we will get to you and hand you over to justice, or you die just like any other dog," he said in an interview with the station.
With conflicting versions of events given by the two sides, the situation on the ground was unclear.
The RSF said the army had attacked it first, while the army said it was fighting the RSF at sites the paramilitaries said they had taken.
A prolonged confrontation between the RSF and the army could plunge Sudan into widespread conflict as it struggles with economic breakdown and tribal violence, and could also derail efforts to move towards elections.
The clashes follow rising tensions between the army and the RSF over the RSF's integration into the military. The disagreement has delayed the signing an internationally backed agreement with political parties on a transition to democracy.
Civilian forces that signed a draft version of that agreement in December called on Saturday for an immediate halt to hostilities by both the army and the RSF, to stop Sudan sliding towards "the precipice of total collapse."
"This is a pivotal moment in the history of our country," they said in a statement. "This is a war that no one will win, and that will destroy our country forever."
The RSF accused the army of carrying out a plot by loyalists of former strongman President Omar Hassan al-Bashir - who was ousted in 2019 - and attempting a coup itself.
Hemedti, the RSF's commander, has been deputy leader of Sudan's ruling Sovereign Council, headed by Burhan, since 2019.
Doctors said a "very large number" of civilians had been killed and injured in the clashes, which extended into several residential neighborhoods across the capital. They only confirmed three deaths as people were sheltering from gunfire.
The army said the Sudanese air force was conducting operations against the RSF. Footage from broadcasters showed a military aircraft in the sky above Khartoum, but Reuters could not independently confirm the material.
Eyewitnesses reported fighting in many areas outside the capital. Those included heavy exchanges of gunfire in Merowe, northern Sudan, eyewitnesses told Reuters.
The RSF shared a video that it said showed Egyptian troops who "surrendered" to them in Merowe. Egypt's military said the troops were in Sudan for exercises with their Sudanese counterparts.
As Cairo sought guarantees for their safety, Hemedti told Sky News Arabia the Egyptians were safe and the RSF would cooperate with Cairo on their return.
The video showed men dressed in army fatigues crouched on the ground and speaking to RSF members in an Egyptian Arabic dialect. Unconfirmed reports by open-source intelligence analysts said several Egyptian Air Force fighter planes and their pilots were captured by the RSF.
Clashes had also erupted between the RSF and army in the Darfur cities of El Fasher and Nyala, eyewitnesses said. The RSF said it had taken control of airports in El Fasher as well as in West Darfur state.
The Sudanese armed forces spokesperson told the Al Jazeera Mubasher television station that the army would respond to any "irresponsible" actions.
Brigadier-General Nabil Abdallah said there was a heavy presence of RSF troops at the TV headquarters in Khartoum.
International powers - the U.S., Russia, Egypt, Saudi Arabia, the United Nations and the European Union - all called for an end to the hostilities.