With the Copa America also to be held this summer, Villa are refusing to release Martinez for a second tournament.
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He told TyC Sports: "I want to participate in the Olympic Games, but it's not just up to me.
"The club blocked me several times. I decided to play the Copa América and then make a decision. With the club not letting any players leave, the situation is not yet 100% clear.
"I always put the Argentine national team first and if I have to fight with my club I will. The most important thing is the Copa America, then the Olympic Games.
"My dream is to win the gold medal."