Barkley, who has already won a senior England cap, has been handed a key role by Martinez and a string of impressive performances have attracted the interest of other clubs.
Barry, on loan from Manchester City, has been able to watch the 19-year-old at close quarters and believes a run in the Everton first team will help his development.
"The manager has already shown he is not frightened to throw the younger players in if he thinks there is a role in the team and he thinks they have enough ability," said Barry.
"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.
"There are so many positives I can throw at him, for such a young player he is playing a very mature game.
"He is strong, quick, has a change of pace and the performances he has shown in the Premier League prove he can deal with the pressure.
"But on the other side the manager has already stated you don't want to put loads of pressure on his shoulders.
"You don't want to build him up too much. 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."