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In a statement released on Monday, the Spanish Football Federation said Costa was "on leave for personal reasons".
Costa and Aspas, who were both part of Spain's World Cup squad over the summer, played against each other on Saturday as Atletico lost 2-0 at Celta.
Costa played the full duration of the LaLiga contest, in which Aspas netted the second goal.
Aspas scored in Spain's last meeting with England, a 2-2 draw at Wembley in November 2016.
The teams face each other at the same venue on Saturday in what will be La Roja's first match under new boss Luis Enrique.
Spain then take on Croatia three days later at Elche's Estadio Martinez Valero.