The plane crashed onto a small island in the White Nile river, close to Juba airport.
Several small farming communities live on the island, and it was not clear if some of the victims were people who were on the ground when the plane hit.
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The radio said up to 40 people may have been killed, adding that airport officials had told them only three passengers had survived.
An AFP reporter at the scene said he could see people trying to search for survivors and carry "several" bodies out of the wreckage.
The main fuselage of the plane had ploughed into thick woodland, with debris scattered around the riverbank in a wide area.
Juba's airport is the busiest in the war-torn country, which is the size of Spain and Portugal combined but with few tarred roads.
The airport hosts regular commercial flights, as well as a constant string of military aircraft and cargo planes delivering aid to remote regions cut off by road.
Civil war broke out in December 2013 when President Salva Kiir accused his former deputy Riek Machar of planning a coup, setting off a cycle of retaliatory killings that has split the poverty-stricken, landlocked country along ethnic lines.
Fighting continues despite an August peace deal, but battles today are far from the capital.
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