There have been a lot of stories about his difficulties, but there has always been a sense that the powers that be, even if not he manager himself, were trying to smooth everything over and protect him. It is notable then that news of the last indiscretion (simply not showing up for training because he was ill and not telling anyone) came directly from the club. It was a clear change in attitude, and while it may simply be an attempt to give him a public kick up the arse, it could also be them indicating that they are done.
As for why Valverde has seen less on board with the player than his superiors are, it's one of those strange ones, maybe kind of like Babel was for us. He has scored some nice goals, he superficially appears to be contributing, and you get a sense that there is a lot more to come. Despite this, there is always a pretty strong sense that they are a better side when someone else is on the pitch instead of him. That is reflected in the manager's reluctant use of him, and probably contributing to the situation not getting better.
In which case I think Klopp would be very reluctant to bring him to Anfield no matter how talented he is.