A crew member and a child on board survived, presidential spokesman Ateny Wek Ateny told Reuters.
Shortly after taking off from Juba airport the plane came down on the banks of the White Nile River, leaving a tail fin and lumps of fuselage strewn in vegetation close to the water.
The plane may have had about 20 people on board, including crew and "probably" 10 to 15 passengers, Ateny said, but added: "We need to confirm how many people were on board."
In addition, he said an unknown number of people were killed on the ground as the Antonov plane crashed near where some fishermen were working. "We don't know the number of people that were killed on the ground," he added.
A police officer, who did not give his name because he was not authorized to speak to the media, told Reuters at the scene that at least 41 people died, but said the number could climb. The Reuters witness said he saw 41 bodies at the site.
Earlier, South Sudanese media had said the cargo plane carried five Russian crew and seven passengers. South Sudan Tribune on Twitter also reported two survivors, one of them a child.