Another 21 people were injured in the incident yesterday at the Los Llanos base, which sent flames and a plume of black smoke billowing into the air, the Spanish Defence Ministry said in a statement.
Rajoy told Spain's Telecinco news program in an interview last night that 11 of the injured were Italian and 10 were French.
Five of the injured with severe burns were transported to a Madrid hospital for treatment and the rest were undergoing treatment in the city of Albacete near the base, the Defence Ministry said.
The two-seat jet was taking off but lost thrust and crashed into an area of the base where other aircraft involved in the NATO exercise were parked, the Spanish Defence Ministry said. At least five jets were damaged and the Italian statement said "numerous" helicopters were damaged.
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NATO Secretary General Jens Stoltenberg said he was "deeply saddened by the crash of a Greek fighter jet at the Los Llanos base in Spain, which has caused many casualties."
The Spanish ministry said the jet that crashed was taking part in a NATO training exercise called the Tactical Leadership Program.
The 10 NATO countries participating in the program are Belgium, Britain, Denmark, France, Germany, Greece, Italy, the Netherlands, Spain and the United States.
According to a US Air Force website, TLP was formed in 1978 by NATO's Central Region air forces to advance their tactical capabilities and produce tactics, techniques and procedures that improve multi-national tactical air operations.
The Los Llanos base is about a two and a half hour drive from Madrid.