The RSF, battling the regular army for more than a year, affirmed in a post on X late Friday "the readiness of its forces to help citizens by opening safe passages to voluntarily leave to other areas of their choosing and to provide protection for them."
El-Fasher, the state capital of North Darfur and once a key hub for humanitarian aid where many had gathered for shelter, has been in the grips of fighting as the RSF seeks to control it.
The paramilitaries called on residents of El-Fasher to "avoid conflict areas and areas likely to be targeted by air forces and not to respond to malicious calls to mobilize residents and drag them into the fires of war".
Sudan has been in the throes of conflict for over a year between the regular army led by de facto ruler Abdel Fattah al-Burhan and the RSF led by his former deputy Mohamed Hamdan Dagalo.
The conflict has killed as many as 15,000 people in the West Darfur state capital of El-Geneina alone, according to United Nations experts.
Medical charity Doctors Without Borders on Wednesday said its hospital in North Darfur had received more than 450 people killed in the fighting since May 10, but noted that the actual death toll was likely much higher.
Tens of thousands of people have died and millions have been displaced since the war broke out in April 2023.
The UN on Friday warned it only had 12 percent of the $2.7 billion it sought in funding for Sudan, warning that "famine is closing in".