Nine were French military personnel, and two were Greek pilots, a senior Spanish defense official told CNN.
The jet crashed on takeoff, hitting other planes on the ground during a NATO training exercise, said the source, who by custom was not named.
Twenty people were injured, according to a statement from the Spanish Defense Ministry. All those hurt are either French or Italian.
NATO Secretary General Jens Stoltenberg said he was "deeply saddened" by the crash.
"This is a tragedy which affects the whole NATO family. I express my heartfelt condolences to the loved ones and the nations of those who lost their lives, and I wish a speedy recovery to the injured," he said in a statement.
"The plane was taking part in an exercise in the framework of NATO's Tactical Leadership Programme, which aims to improve multinational cooperation in air operations," Stoltenberg said.
The NATO flight training course at the Spanish base has 750 participants, including pilots, ground crews and other support, from nine countries: the United States, United Kingdom, France, Germany, Italy, Spain, Belgium, Denmark and Greece, the Spanish air force reported on its website last week.
It was expected to be a four-week program, from January 19 to February 13, with training to prepare pilots for tactical missions, including nighttime missions, the site said.