Prime Minister Mariano Rajoy said there had been "eight to 10 people on board."
Speaking on Tenerife, one of the Canary Islands, where he was attending a campaign meeting, Rajoy sent his condolences to the victims, Spanish news agencies reported.
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 delivery delays, went into service.
The aircraft has been promoted by Airbus as a possible replacement for New Zealand's aging Hercules aircraft, with the planemaker taking out a series of adverts in the Herald for the A400M plane.
-AAP