The 20-year-old joined the Championship club on a one-month loan deal in September and he has impressed during his stint with the Owls thus far.
It is understood that the youngster's loan spell will end this weekend, but Jones remains hopeful that he can agree a long-term deal.
"Hopefully we can keep Ross. I think they want us to have him because the kid hasn't done anything wrong," he told the Sheffield Star.
"Ross is a kid and going to make mistakes, but he's some player when he drives on and plays the way he does."
Barkley has made 10 appearances for Everton.