The 19-year-old is due to end a month at Elland Road next week - but Leeds manager Neil Warnock will seek talks with Toffees boss David Moyes with a view to extending his stay.
Warnock said: "I'll be talking to David Moyes over the weekend. He said he'd let him come for a few games and we'd have a chat nearer the time.
"You just don't know with loan situations but we'll see what develops. He's still in my thoughts, very much so."
Barkley has already started three games for the Championship club and has made the squad for Saturday's trip to Molineux to face Wolverhampton.