"There are at least three dead and two seriously injured," a spokesman for the emergency services told AFP.
The plane crashed in a non-residential area around a mile north of Seville airport, which was closed to traffic for about an hour.
Airport authority AENA said on Twitter that the airport's firefighters were at the scene of the crash, "beyond the airport's perimeter."
Prime Minister Mariano Rajoy said there had been "eight to 10 people on board".
Aviation sources confirmed the plane was one of Airbus' new A400M troop transporters, which are assembled at a factory in Seville.
Airbus Defence and Space, the Airbus division responsible for military aircraft, said the plane was destined for Turkey.
The group sent a team of experts to the crash scene.
If confirmed as an accident, it would be the first since the aircraft, which was fraught by development and shipping delays, went into service.
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