I'm not sure that's the case. This feels very much like part of a contract negotiation. He absolutely isn't going to move to another PL club. He's not stupid and he knows that it will not enhance his brand to play an extra 10 games a season but win the square root of fuck all. It's a non starter.
I agree that if Madrid were to come knocking with a big offer he'd probably jump at it, but that has always been the case with him. The rest of the so called interested parties - well Barcelona haven't got a pot to piss in, and Bayern are absolutely going to land Mane.
This is just my opinion, but I think ultimately Sterling is smart and aware enough about how good he's got it at City. There's little point in running his contract down because the situation next summer will be the same as this. There is only really one club that he would want to go to and if they aren't interested this summer they won't be interested next when he's a year older.
I think he ends up signing a new deal.