Presidential spokesman Ateny Wek Ateny said around 20 people were aboard the plane.
Only one crew member and a child survived.
An unknown number of people were also killed on the ground when the plane came down near a group of fishermen.
“We don’t know the number of people that were killed on the ground,” Mr Ateny said.
He added that the aircraft was a Russian-made cargo plane bound for the Paloch oil fields in Upper Nile state.
The Russian news agency Interfax, citing a source, said preliminary findings indicate the plane did not belong to any Russian airlines.
A police officer told Reuters at least 41 people died, adding that the death toll is likely to rise.