Army chief Fattah al-Burhan agreed to facilitate and secure their evacuation "in the coming hours", it said, according to BBC.
He is locked in a bitter power struggle with the leader of a rival paramilitary faction, the Rapid Support Forces.
Hundreds of people have been killed in a week of fighting across the country.
Previous plans to evacuate foreign nationals have not been implemented because of safety fears.
A statement from the army said British, US, French, and Chinese nationals and diplomats would be evacuated by air on board military transport planes from the capital, Khartoum.
Saudi Arabia has also announced it is arranging the evacuation of Saudi citizens and nationals of "brotherly" countries. The Sudanese army said Saudi Arabia's diplomatic mission had already been evacuated.
The conflict has entered its second week despite both sides - the army and the RSF - agreeing to a three-day ceasefire to mark the Muslim holiday of Eid al-Fitr, starting from Friday.
Khartoum's international airport has been closed due to the violence, with foreign embassies - including the US and UK - unable to bring their citizens home.
Street battles erupted in the capital, Khartoum, on 15 April after disagreements emerged between the leaders of both sides - General Burhan and the RSF's Mohamed Hamdan "Hemedti" Dagalo - over how Sudan should be run.
They both held top positions in Sudan's current military government, formed after the 2019 coup that ousted long-time leader Omar al-Bashir.
They were supposed to merge their forces but the RSF resisted this change, mobilising its troops which escalated into full-scale fighting last week.
MNA/PR