I don't think Koeman doubts Barkley's talent, he's been highly creative this season.
He's asked Barkley, as well as others, to be more productive. For Barkley it's goals and assists, and he hasn't got those up to a quality level.
These days being given a season to impress is actually a long time, on top of Barkley being a regular first team player for a few years now.
Whether he can manage Barkley or not, I'm not too bothered. If he can't, then let him get a player he can manage and that can meet the managers demands more proficiently. There's nothing wrong with that.