The paramilitary Rapid Support Forces (RSF) says it has taken control of at least three airports, the army chief's residence and the presidential palace, APA reports citing BBC News.
The army denies this and says its air force is attacking paramilitary bases.
People are attempting to shelter from the fighting.
An eyewitness said residents had not been expecting the clashes and many had been caught in transit. with bridges and roads closed and many schools in lockdown.
US Ambassador John Godfrey said he "woke up to the deeply disturbing sounds of gunfire and fighting. I am currently sheltering in place with the Embassy team, as Sudanese throughout Khartoum and elsewhere are doing". He urged senior military leaders to stop the fighting.
There are also reports of clashes at the state TV station.
Earlier the RSF had said that one of its camps in the south of Khartoum had been attacked.
For its part, the army has said that RSF fighters have been trying to seize the military headquarters.