The war in the African country is approaching its third month with no end in sight. The casualties were reported by the Al Naw hospital in Omdurman.
The army, led by Gen Abdel Fattah Al Burhan, is reportedly trying to cut supply routes used by its paramilitary rivals, the Rapid Support Forces, led by Gen Mohamed Dagalo.
Violence between the two sides erupted in mid-April as the generals vied for control following a military coup that overturned an attempted power-sharing agreement with civilian-led political parties.
The war has now reignited simmering ethnic conflict across the country, including in Darfur, the scene of mass killings by government-linked militias about 20 years ago.
On Tuesday, the army launched air strikes and heavy artillery salvos, and there were ground battles in several parts of Omdurman, witnesses said.
The RSF quickly took control of large parts of the capital and has brought in extra fighters from Darfur and Kordofan as the conflict has deepened.
It transferred the reinforcements across bridges from Omdurman to Bahri and Khartoum, the other two cities that make up the wider capital across the confluence of the River Nile.
Residents said Tuesday's clashes in Omdurman were the heaviest in weeks.
As the army tried to gain ground, it was also fending off an RSF attack against a police base, they said.
“There has been very heavy bombardment for hours – air strikes, artillery and bullets. It is the first time for us that there have been continuous strikes at this level from every direction,” said Manahel Abbas, 33, a resident of Omdurman's Al Thawra district.
Saudi Arabia and the US brokered several ceasefire deals during talks in Jeddah. The talks were suspended last month after both sides breached the truces.
In a move that could escalate conflict in western Sudan, tribal leaders in South Darfur declared their allegiance to the RSF on Monday.
The RSF originated in Arab militias that helped to crush a rebellion in Darfur after 2003, before developing into a national and officially recognised force.