The club has form for using this form of spin. I would imagine that a lot of people have been talking to him, telling him to keep his head down and focus on his football and that is exactly what he has done. In that circumstance, it's difficult, but he can concentrate on his performance and try to zone everything else out.
I think that the club has bided it's time until the end of the window to try and extract the maximum price from Chelsea, just as they did with Fellaini's transfer to United. The player will have been told that he will need to hand in a transfer request though, as this takes the heat off Kenwright and Martinez. Stones, feeling like he has no option to refuse, obliges. Handily, it also saves us from having to cough up any loyalty payments.
Even if the club aren't behind it, Stones has obviously wrestled with this decision. Everton have great promise and he obviously feels settled at the club but Chelsea will be playing in the Champions League regularly and competing for cups and titles. He would get the opportunity to work with perhaps the best manager in the world and the chance to realise his full potential.
I remember reading about Gerrard saying that he had a panic attack when deciding about whether to leave Liverpool for Chelsea and that he called the club doctor and his brother for reassurance. It can't be an easy situation and I'm just thankful that he's handled it better than Stirling.