The 19-year-old midfielder put in an impressive performance during the Merseyside club's 3-1 pre-season victory over Blackburn Rovers at Ewood Park on Saturday.
Barkley had spells on loan at Championship clubs Sheffield Wednesday and Leeds United during last term but did feature for Everton towards the back end of the season.
And Martinez is excited to see the youngster develop his talent at the club.
"He is something special," he told The Liverpool Echo.
"In football when you describe a player, there are not many better descriptions than to say he's an incredible football diamond.
"He's still a young man, and needs to learn from the players around him. But I'm excited to have that process with him. He has got incredible talent.
"I am really pleased with his attitude, the way he listens and wants to learn. And when he is in full flow, he can be something special."
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