The 18-year-old has scored four goals in 13 appearances for the Championship club since joining in September and he has revealed his delight at his current form.
"I've followed the career of Dave Jones for a long time, from when he was at Cardiff. He's a Scouser from the same area as me so I've always taken an interest in his career," Barkley told the club's official website.
"I was made up when I found out he was interested in bringing me to Wednesday. I've really enjoyed my time here so far. I feel as if I have made a lot of progress and that's the most important thing. The manager and the coaching staff talk to me all the time and that's a great help. I'm playing regular football at a good level in the Championship so I'm really pleased with how things are going.
"I'm taking things one step at a time and we will see what happens from here. I see my future in the long term with Everton, but right now, I'm enjoying my football and the experience I'm getting with Sheffield Wednesday."
Barkley has been strongly linked with a move to Manchester United.