The 18-year-old, voted Owls fans' player of the month for October, has made a big impact since arriving on loan in mid-September, scoring his fourth goal in 13 appearances in the recent win over Peterborough.
Barkley, who has credited Owls boss Dave Jones and his coaching staff with his progress at Hillsborough, told Wednesday's official website: "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.
"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."