The reports come just hours after the RSF (Rapid Support Forces) stated that they have taken over the army's headquarters in the city. The message has not been confirmed by independent parties.
As a result of intense fighting, access to the region is extremely limited.
In a statement, the militant movement states that they "have expanded their control over the city of al-Fashir at the expense of mercenaries and militias", by which they mean the country's army.
UN appeals
Late on Sunday evening, the UN urges the combatants to allow civilians to leave the city in a safe manner.
"In a situation where fighters are getting further and further into the city and escape routes are being cut off, there are hundreds of thousands of terrified civilians who are trapped - they are being shelled, starved and lack access to food, healthcare and security", says the UN's humanitarian chief Tom Fletcher in a statement.
The city of al-Fashir (El-Fasher) has been under siege for more than 18 months, with dreadful consequences for the hundreds of thousands of civilians who remain. The RSF, which is at war with Sudan's regular army, has long tried to take the city.
Accused of serious war crimes
Both parties have been accused of serious war crimes.
A local militia allied with the army states, however, in a statement that residents are still "resisting terrorist militias."
If the RSF's reports are true, it means a victory for the notorious militia, in a civil war that has killed tens of thousands of people, driven around 12 million to flee and caused what is called the world's worst humanitarian situation.
