It depends on his attitude between now and the end of august. Without knowing exactly what's going on behind the scenes, it's hard to say for sure. However I do think they would call his bluff till the end by telling him PSG or nothing. He may be sold but I just can't see them selling to City. Not any more.
Why not City? Everything points to Sanchez saying it's City or I'm off next summer for free. Who has more to lose in that situation? I would say Arsenal has much much more to lose in that situation, I don't think Sanchez is bluffing, it's a win-win for him. He gets his move to City or he leaves for free next summer and becomes the most wanted free agent in history, guaranteeing himself a massive signing on bonus wherever he decides to go. I believe almost every top club will make an offer to Sanchez to try to convince him to join. He could sign a pre contract with a foreign club in January, there will be massive demand for his signature already in October/November, maybe even before that.