Galloway has cemented the starting left-back position in the absence of the injured incumbent Leighton Baines and Martinez believes he has earned the right for a new contract.
"Brendan Galloway is a first-team player, he is not a young player any more," Martínez said of the 19-year-old.
"With what he has done this season he has earned the right to be considered a starter.
I don't consider Brendan as a teenager who is just filling a role in the first team. He has earned the right of being an important player in the first team and that's the way I see it."
"I would like him to get a different contract. He has got three years left [after this season] and he's in line for a different one. We have been talking with his agent. We have just exchanged ideas at the moment and we'll see where we go with that."