The 19-year-old midfielder has impressed at Goodison Park this season after being thrown into the first team by manager Roberto Martinez.
Barry, who is currently on loan from Manchester City, is convinced that Barkley should remain with the Toffees if he wants to get better.
"Ross is reaping the benefits at the moment and the more he plays the more confident he will get and the more he will improve," Sky Sports News quotes him as saying.
"I think Everton is the perfect club to be learning his trade and the more he plays the more he learns and he will keep improving while keeping his feet on the ground, which will be ideal for him."
Barkley has scored two goals in seven games for Everton so far this season.