The Greek F-16 jet lost power as it took off from its base near Albacete near Madrid said the defence ministry in a statement.
The Falcon then hit two Italian AMX planes and three French jets - a Mirage and two Alfa jets.
The two Greek pilots of the crashed plane, both captains, died in the accident with nine French military - three captains and a lieutenant.
21 people were injured in the crash: 11 Italians and 10 French people.
Amongst the French nationals, five of them suffered major burns, one is in serious condition and another one has been placed in an induced coma.
French president François Hollande has expressed his "deep respect for the commitment" of the airmen who were preparing for air force missions to fight "against terrorist groups" in Iraq and Sahel.
The base has been housing training for Nato pilots since 2010 and 10 countries take part in the programme: Belgium, Britain, Denmark, France, Germany, Greece, Italy, the Netherlands, Spain and the United States.