Why would he go there and win less than he will at City? That's a genuine question. I don't see how it enhances his rep to play an extra 10 games a season as I said earlier, and I don't see how they would pay him more than City do. Not from his side, nor from Chelsea's side does the transfer make sense.
Also for the record, I absolutely don't believe any of this is about money. Sterling is on top whack already at City. His next deal will be a pay rise without question, but that isn't going to decide whether he signs it or not. It'll be football related, and in that sense I don't see how anyone outside Madrid tempts him.