Martinez told Radio Continental: "I have shown the club that I can play for them and when I return next season I need to have more games. That is the only way I will stay at Arsenal.
"My target is to consolidate my place with Arsenal and to reach the Argentinian national team as number one.
"I won't stop until I get that and if I don't play for Arsenal, then clearly I would move on."
The goalkeeper then burst into tears in an interview after winning the FA Cup final when reminded about his origins.
Leno, Martinez's main competitor for the Arsenal number shirt, was already back in training before the Wembley win but was not fit enough to make the matchday squad.