A video posted by the RSF showed its soldiers cheering in front of a sign for the army's Sixth Infantry base.
Reuters was able to verify the location but not the date.
The army did not immediately give a statement on its current position.
Capture of al-Fashir would be a significant political victory for the RSF that could hasten a physical split of the country, by enabling the paramilitary group to consolidate its control over the vast Darfur region which it has claimed as the base for a parallel government established this year.
Yet some activists have long warned that an RSF takeover of the city would also lead to ethnic attacks, as seen after the capture of the Zamzam camp to the south.
The RSF had besieged the city, capital of North Darfur state, for the past 18 months as it fights the army and allied former rebels and local fighters.
It has targeted civilians in frequent drone and artillery strikes while the siege has led to starvation among the 250,000 people who remain in the west of the city.
Reuters was unable to reach residents of al-Fashir, who rely on Starlink terminals to access the internet given a long-term telecom blackout.
Last week, the RSF said it was facilitating the exit of civilians and surrendered fighters from al-Fashir but those who have left have reported robberies, sexual assaults and killings by RSF soldiers on the road.
A United Nations-mandated mission said last month the RSF had committed multiple crimes against humanity in the al-Fashir siege.
The army has also been accused of war crimes.
The Popular Resistance, local fighters allied with the army, said they were continuing to fight the RSF and that the takeover of the army base did not mean the takeover of the city.
The war between the Sudanese army and the Rapid Support Forces began in April 2023 when the two forces, previously sharing power, fell out over plans to integrate their forces during a transition to democracy.
The fighting has displaced millions, plunged half of Sudan into hunger and spread disease throughout the country.
Over the weekend, the United States convened Emirati, Egyptian and Saudi Arabian officials to discuss a possible peace plan.
Sudan's foreign ministry said officials were in Washington DC for bilateral talks.
However, the army-led Sovereign Council denied reports that representatives of both forces were engaging in indirect talks.
