Can see him being kept on for this season at least, he's not going to be on the radar of the clubs that he thinks want him and the clubs that do want him won't be able to meet out valuation or match his footballing ambitions. I know we don't ever get the full picture of why a player leaves a club until they release an autobiography about 20 years after the events but it does seem very daft to want to leave Everton at this moment in time. He'll get plenty of game time if he doesn't act like a bell and pull his socks up because there's no more free riding under Koeman plus he'll be able to work with Rooney who is probably the best currently playing type of player that Barkley should model his game on: a deep lying forward/advanced midfielder.
Depending on how bellish he's going to act I would consider pushing him into the striker role and see how he does up there. He's got the physicality and talent plus it lets him not have to think too hard as the much cleverer players behind him will do his thinking for him so all he has to worry about is putting the ball in the net. Depending on how the hunt for a striker goes it may be a decent option than relying on Rooney and Calvert-Lewin for the entire season considering his favoured position is already well covered.
Either way would rather he stayed but only if he's going to get his head down and actually do some work rather than throwing his toys out of the pram because an actual manager is running the show now and wants to see his players put a shift in. If he's too immature to let bygones be bygones then flog him to the highest bidder and use the money as best we can to patch other parts of the team. After all, football is a business these days and if an injury prone so called boyhood blue is running his contract down and not wanting to make use of his talent here then we shouldn't give him the chance to move elsewhere for a pittance; we wouldn't give another player who doesn't have those ties to us the chance so why Barkley?