The RSF did not comment, while a source said the paramilitary had nothing to do with the outages.
Many Sudanese reported being unable to reach family members, compounding the impacts of ongoing fighting between the RSF and the Sudanese Armed Forces, SAF, that has forced more than 7.5 million people from their homes and prompted warnings of famine.
Millions rely on online payments for food and other essentials, now inaccessible due to the network disruption.
Netblocks, an internet observer, shared data on X on Sunday that showed connectivity for two major providers, South African-owned MTN Sudan and state-owned Sudani, falling to zero and close to zero beginning on Friday.
According to state news agency SUNA and the two sources, the RSF shut the two providers' connections down as it demanded the restoration of network outages in the western region of Darfur, which it largely controls.
The cause of outages in Darfur was not immediately clear. MTN and Sudani did not respond to a request for comment.
A RSF source blamed the Darfur outages on the army, saying it had ordered months-long outages there and other areas that had not drawn similar outside concern.
The source did not say who was responsible for the outages in other parts of the country but said the paramilitary group was not behind them.
RSF controls most of the capital Khartoum, including telecoms companies' facilities. The two telecoms sector sources said the RSF was able to shut down the networks without causing permanent damage.
The RSF has also forced Sudan's third main provider, Kuwaiti-owned Zain Sudan, to stop service in River Nile state and Port Sudan city, both controlled by the army, the sources and SUNA said.
In a statement on Monday posted on Facebook, Zain said that its employees "are working under very difficult, harsh, and dangerous circumstances, and [Zain] would like to note that the current network outage is due to circumstances outside of its control."
A representative for Sudani declined to comment, but on Saturday the company said on Facebook it was working to restore service.
MTN parent company MTN Group said in a statement to Reuters on Monday that the outage arose from the ongoing conflict.
"MTN Sudan is actively engaging with relevant stakeholders to minimize the duration of this interruption," it said.
Network towers, power lines and other infrastructure have been damaged in fighting across Sudan since the conflict broke out in April over plans to integrate the RSF and the army as part of a political transition towards elections.
The war has made Sudan the world's biggest internal displacement crisis and both sides have been accused of war crimes.
Medical charity MSF warned on Monday that at least one child was dying every two hours in Zamzam, a large camp for displaced people in Darfur.