Martinez is currently third choice at the Emirates behind Petr Cech and David Ospina.
But the Argentine, who spent time at Wolves last season, signed a new long-term contract with Arsenal last November.
And the 24-year-old has now claimed Wenger feels he is the future of the club.
Emiliano Martinez has revealed Arsene Wenger believes he is the future at Arsenal
"It's amazing that Arsene Wenger, one of the best coaches in the world, believes I'm the future of Arsenal," Martinez told Goal.
"Just as amazing as having my contract renewed for six more years at the club.
"Wenger is very well-respected. He's not only the head coach here, he also built Arsenal. He helped buy the stadium.
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Emiliano Martinez is currently third choice at Arsenal
"Everybody calls him 'boss', from the kit manager to the owner of the club. He seems distant, but he's always available to talk. It's impressive.
"Cech is a great goalkeeper, he has a lot of stories, he's in the top 10 in the world. We get along, he also says I'm the future of the club.
"Also Ospina, he's such a professional. He's a great guy and friend. I usually have dinner at his house, and I learn a lot from him.
"It's an honour that the coach gives me the possibility to be the goalkeeper in a tournament [the EFL Cup], when he has those two."