The Owls are hoping to hold onto the young Toffees man beyond his current loan deal, which is set to expire after the club's Yorkshire derby with Leeds United
The Championship club signed the England Under-19 player on a short-term deal in September, but are hoping the Toffees will allow 18-year-old Barkley to stay beyond his last game, which is rumoured to be their encounter with Leeds.
In six appearances, Barkley has scored once for Wednesday and Jones hopes that the talented youngster will still be able to contribute more.
"Hopefully we can keep Ross," he told the Sheffield Star.
"I think they [Everton] want us to have him because the kid hasn't done anything wrong.
"Ross is a kid and is going to make mistakes, but he's some player when he drives on and plays the way he does."
Having joined Everton aged 11, Barkley has long-been tipped as one of England's brightest prospects. But after recovering from a serious leg injury though in 2010, he has since struggled to break into the Toffees' first team.