Stones made his Everton debut in last week's Capital One Cup defeat of League One Stevenage having joined from Barnsley in January.
The England Under-21 international is now keen to progress at Goodison Park and sees Barkley as a role model.
"Of course it is a big inspiration for me when Ross is making the first team and getting picked for England," said Stones.
"He is hitting the ground running for Everton. I really want to do the same as Ross.
"I am always trying to push for a first-team position, as is everyone in the dressing room. It is healthy competition.
"I want to cement a spot in the first team, to keep pushing as hard as I can all season until I get there. That is my big aim."
Stones is delighted to have been given the chance to play for Everton in some high-profile pre-season matches and is keen to establish himself in Roberto Martinez's plans.
"To get that first game in pre-season was a big help for me, getting on and playing against Juventus, and then the likes of Real Madrid and Valencia," he added.
"It was a great experience for me and made me feel like I had hit the ground running.
"Coming into such a big dressing room, the lads have taken to me so well and I am so thankful for that.
"Hopefully I can get a few more games now."