The clashes began at dawn, several residents reported, in what appeared to be the army's first major offensive in months to regain parts of the capital controlled by the rival paramilitary Rapid Support Forces.
It comes the same day Sudan's de facto leader, army chief Abdel Fattah al-Burhan, addressed the UN General Assembly in New York.
On the eve of his address, UN chief Antonio Guterres voiced concern to Burhan "about the escalation of the conflict in the Sudan", the United Nations said.
Burhan called in his address for the "rebel militia to be designated as a terrorist group" and accused the RSF of obstructing peace efforts.
On the ground, Sudanese army forces were "waging fierce fighting against the rebel militia inside Khartoum", a source in the military told AFP, referring to the RSF.
The source, requesting anonymity because they were not authorised to speak to the media, said army forces had crossed three key bridges over the Nile -- which had separated parts of the capital held by the army from those under RSF control.
- 'Political cover' -
Since April 2023, when war broke out between Burhan's Sudanese Armed Forces and his former deputy, RSF commander Mohamed Hamdan Daglo, the paramilitaries had largely pushed the army out of Khartoum.
In its last major offensive in February, the army regained much of Omdurman, the capital's twin city across the Nile and part of greater Khartoum.
The government loyal to the army is based in Port Sudan on the Red Sea coast, where the army has retained control.
The RSF meanwhile has taken control of nearly all of the vast western region of Darfur, rampaged through the agricultural heartland of central Sudan and pushed into the army-controlled southeast.
In his address at the UN, Burhan accused "countries in the region" of providing the RSF with "financial support, mercenaries and political cover".
The army chief did not name the countries, but his government has repeatedly accused the United Arab Emirates of funnelling arms to the paramilitaries from neighbouring Chad in violation of the arms embargo on Darfur.
United Nations experts last year found the accusations "credible", and diplomats say the United States has in private pressed the UAE over its support to the RSF.
- 'Intense' shelling -
In Omdurman, a witness said the sound of bombs, which had "jolted them from sleep" at dawn, continued through the day.
Several residents of Omdurman reported "intense artillery shelling" falling on residential buildings while military warplanes flew overhead.
"In our neighbourhood alone, three people have been killed today," one Omdurman resident told AFP, requesting anonymity for fear of reprisal.
Amid an effective communications blackout and a crippled healthcare system, it was not possible to verify any casualty toll.
Since the war began, much of its worst fighting has been in densely populated areas and both sides have been accused of war crimes, including indiscriminately bombing residential areas.
The war has already killed tens of thousands of people, with the World Health Organization saying at least 20,000 people have been killed, but some estimates of up to 150,000.
It has also created the world's largest displacement crisis, the UN says.
More than 10 million people -- around a fifth of Sudan's population -- have been forced from their homes, according to UN figures.
Famine has been declared in Zamzam refugee camp in Darfur near the city of El-Fasher, where the RSF last weekend launched a large-scale offensive after months of siege.
El-Fasher is the only one of five state capitals in Darfur not yet in RSF hands.
In his meeting with Burhan, Guterres said the war "risks a regional spillover," the UN said.
"People in Sudan have endured 17 months of hell, and the suffering continues to grow," top UN relief official Joyce Msuya said separately.
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