Battles have taken place throughout the vast country and there are fears of regional spillover.
The conflict has seen air strikes, artillery and heavy gunfire.
Those compelled to venture out face queues for bread and petrol at outlets that are not shuttered. Residents are also dealing with power outages.
U.S. Secretary of State Antony Blinken said Tuesday he had spoken with the two generals and "underscored the urgent need for a ceasefire."
"Too many civilian lives have already been lost," Blinken tweeted, adding he had "stressed the importance of ensuring the safety of diplomatic personnel and aid workers."
Following the call, Mohamed Hamdan Dagalo said in a tweet that the pair had "discussed pressing issues in Sudan," adding he was grateful for Blinken and the US's "commitment to restoring stability in Sudan."
Blinken made his calls while in Japan for a meeting of G7 foreign ministers, who also urged the warring sides to "end hostilities immediately" and "ensure the safety of all civilians."
In a statement, Blinken and other G7 foreign ministers meeting in Japan warned the fighting "threatens the security and safety of Sudanese civilians and undermines efforts to restore Sudan's democratic transition".
But despite their call for the warring parties to "end hostilities immediately", loud explosions were heard on Tuesday morning in Khartoum, where militiamen in turbans and fatigues roamed the streets.
Terrified residents of the capital are spending the last and holiest days of Ramadan watching from their windows as tanks roll through the streets, buildings shake and smoke from fires triggered by the fighting hangs in the air.
"Bombardments usually start around 4:00 am and they continue for a few hours, but today they haven't stopped," said Khartoum resident Dallia Mohamed Abdelmoniem.
"We haven't slept in the past four days," she said, adding her family had been staying indoors "trying to keep our sanity intact."
Volker Perthes, the head of the United Nations mission to Sudan, told reporters after a Security Council meeting that "It's a very fluid situation so it's very difficult to say where the balance is shifting to."
On Monday, UN Secretary-General Antonio Guterres again urged Sudan's warring parties to "immediately cease hostilities". He warned that further escalation "could be devastating for the country and the region."
The European Union's ambassador to Sudan was attacked in his home in Khartoum on Monday, the bloc's top diplomat Josep Borrell said. A spokesperson told AFP the veteran diplomat was "OK" following the assault.
Medics in Sudan had earlier given a death toll of nearly 100 civilians and "dozens" of fighters from both sides, but the number of casualties was thought to be far higher, with many wounded unable to reach hospitals.
The official doctors' union warned fighting had "heavily damaged" multiple hospitals in Khartoum and other cities, with some completely "out of service."
A number of organizations have temporarily suspended operations in the country, where one-third of the population needs aid.
"This renewed fighting only aggravates what was already a fragile situation, forcing U.N. agencies and our humanitarian partners to temporarily shutter many of our more than 250 programs across Sudan," said UN emergency relief coordinator Martin Griffiths.
In the western region of Darfur, international medical aid organization Doctors Without Borders (MSF) reported receiving 183 wounded patients at the only hospital in El Fasher still operating in North Darfur state, 25 of whom it said had died.
"The majority of the wounded are civilians who were caught in the crossfire -- among them are many children," MSF's Cyrus Paye said.
Three U.N. World Food Program staff were also among those killed on Saturday in Darfur, where humanitarian missions have had medical and other supplies looted, according to Save the Children and MSF.
Influential northern neighbour Egypt said it had discussed with Saudi Arabia, South Sudan and Djibouti - all close allies of Sudan - "the need to make every effort to preserve stability and safety".
President Abdel Fattah al-Sisi called on the two warring parties to "return to the negotiating table" and said he was working on the return of Egyptian military "trainers" captured Saturday at an air base by RSF forces.