Late on Sunday, residents reported intense fighting across the three cities that make up the nation's wider capital - Khartoum, Omdurman and Bahri - and smoke could be seen rising from several areas early on Monday.
In Khartoum East district, RSF troops who have spread out in neighbourhoods across the capital were in full control and were looting extensively, said Waleed Adam, a resident of the area.
Residents have also reported widespread looting and insecurity in the area.
There was no immediate comment from the army, which had denied on Sunday that the RSF had taken the town.
In El Obeid, a city 360 km (220 miles) southwest of Khartoum and on a key route from the capital to Darfur, residents reported large deployments of RSF forces and the closure of some roads.
Persons:
Mohamed Nureldin Abdallah, Mohamed Saleh, Waleed Adam, El, Khalid Abdelaziz, Adam Makary, Aidan Lewis, Sriraj
Organizations:
Rapid Support Forces, REUTERS, U.S, Reuters, North Darfur State, Thomson
Locations:
Khartoum North, Sudan, Sudan Khartoum, Darfur, DUBAI, Khartoum, Saudi Arabia, Omdurman, Bahri, Khartoum East district, DARFUR, Sudan's, Kutum, North Darfur, El Obeid, Dubai, Cairo