The flare-up was reported in several areas of the capital city on Sunday, after the truce ended amid the breakdown of bilateral talks in Jeddah, where Saudi Arabia and the United States have been mediating negotiations between the warring factions. The ceasefire started on May 22 and expired on Saturday evening.
Hundreds of people have died since the army and the RSF started to fight as a result of a power struggle in mid-April.
The conflict has caused a major humanitarian crisis in which more than 1.2 million have been displaced, including some 400,000, who have fled to neighboring countries. It also threatens to destabilize the region as a whole.
Fighting in the capital has also led to widespread damage and looting, a collapse in health services, power and water cuts, and dwindling food supplies.
"In southern Khartoum, we are living in terror of violent bombardment, the sound of anti-aircraft guns and power cuts," 34-year-old Sara Hassan told Reuters by phone, adding, "We are in real hell."
Meanwhile. Sudan's antiquities authorities said RSF fighters have withdrawn from the national museum in central Khartoum after on Saturday, they released a video filmed inside the grounds of the museum.
Separately, a new outbreak of violence has been reported in Darfur, in the far west of Sudan, which has left at least 40 people dead and dozens more wounded.
Witnesses reported heavy fighting on Friday and Saturday in Kutum, which is one of the main towns and a commercial hub in North Darfur. The Sudanese army has denied claims by the RSF about taking over Kutum.
RHM/PressTV