Fighting in the city between the army led by General Abdel Fattah al-Burhan and the paramilitary Rapid Support Forces (RSF) led by General Mohamed Hamdan Daglo is now concentrated around military bases.
At the same time in Sudan's west, the conflict is worsening to "alarming levels" in Darfur, the United Nations warned.
Since the war erupted on April 15, the RSF has established bases in residential neighbourhoods of the capital while the army has struggled to gain a foothold on the ground despite its air superiority.
As the RSF fights to seize all of Khartoum, millions of people are still holed up despite being caught in the crossfire without electricity and water in oppressive heat.
Late Sunday, the RSF said it had seized the headquarters, on Khartoum's southern edge, of the paramilitary Central Reserve police, sanctioned last year by Washington for rights abuses.
On Tuesday the RSF attacked army bases in central, northern and southern Khartoum, witnesses said.
Mawaheb Omar, a mother of four who has refused to abandon her home, told AFP that Eid, normally a major event in Sudan, will be "miserable and tasteless", as she cannot even buy mutton, a usual part of the feast.
Burhan took to state television on Tuesday to urge "all the young people of the country, and all those who can defend it, not to hesitate to do so (...) or to join the military units".
The United States, Norway and Britain, known as the Troika, on Tuesday condemned "widespread human rights violations, conflict-related sexual violence, and targeted ethnic violence in Darfur, mostly attributed to soldiers of the Rapid Support Forces and allied militias".
RSF are descended from Janjaweed militia unleashed by Khartoum in response to a rebel uprising in Darfur in 2003, leading to war crimes charges.
In the current fighting the RSF has been accused of looting humanitarian supplies, factories and houses abandoned by those displaced by the fighting or taken by force.
Daglo responded to these accusations on Tuesday in an audio recording posted online.
"The RSF will take swift and strict action" against those in its ranks who have carried out such abuses, he said.
The RSF had said Monday evening that it was beginning to try some of its "undisciplined" members, and announced the release of "100 prisoners of war" from the army.
Since the beginning of the conflict, both sides have regularly announced prisoner swaps through the Red Cross, without ever giving the exact number of those captured.
Daglo, a former Darfur militia chief, also warned against "plunging into civil war".
The UN and African blocs have warned of an "ethnic dimension" to the conflict in Darfur, where on Tuesday Raouf Mazou, the UN refugee agency's assistant high commissioner for operations, told a briefing in Geneva there is a "worsening situation" in West Darfur state.
"According to reports from colleagues on the ground, the conflict has reached alarming levels, making it virtually impossible to deliver life-saving aid to the affected populations," he said.
Elsewhere in the country, new fronts have opened against the army from a local rebel group in South Kordofan state, south of the capital, as well as in Blue Nile state on the border with Ethiopia.
In South Kordofan, authorities have decreed a night-time curfew to curb the violence.
The Troika expressed "deep concern" about the fighting in Blue Nile and South Kordofan, as well as Darfur, that "risked further broadening the conflict".
Hundreds of civilians have fled over the border to Ethiopia because of the fighting reported around Kurmuk in Blue Nile, the UN said.
This adds to the ever-increasing number, now almost 645,000 people, who have fled to neighbouring countries, mostly Egypt and Chad, according to the latest International Organization for Migration data. Around 2.2 million people have been displaced within Sudan, the agency said.
A record 25 million people in Sudan need humanitarian aid and protection, the UN says.