Look, every team in baseball wants Price. There are, maybe, three teams where isn't the number 1. But can they afford the extension. Not really the Cards style.
Still see the Rangers as the front runner. Honestly don't see anyone coming close to being able to offer what they can. Dodgers could be a player. Cubs and Mariners all have deep farm systems but I don't see it happening.
But if the Rays want a pitcher in return, I'd cough up Detwiler or Zimmermann in a heartbeat.
Of course every team wants Price, just like every team wants a Trout, Harper or Puig. Obviously you need to evaluate the farm system on what can be given up to get Price, and then if said team can sign him to an extension. Even though an extension isn't necessarily needed, I think it's a forgone conclusion that whoever is going to trade what it will take to get Price would want to extend his contract. That being said:
Angels - Don't have the prospects to trade.
Athletics - Don't have the money to extend.
Mariners - Have both prospects and money.
Astros - Re-building franchise.
Rangers - Have both prospects and money.
White Sox - Don't have the prospects to trade, unless Sale.
Tigers - Don't have the prospects to trade.
Royals - Don't have the money to extend.
Indians - Don't have the prospects OR the money to extend.
Twins - Don't have the money to extend, also rebuilding franchise.
Yankees - Don't have the prospects to trade.
Red Sox - Have both prospects and money.
Orioles - Don't have either prospects or money.
Blue Jays - Don't have the prospects to trade.
Nationals - Might have the prospects, definitely have the money.
Braves - Don't have the prospects, might have the money.
Phillies - Don't have the prospects.
Mets - Have both prospects and money.
Marlins - Have prospects but money is questionable, also re-building.
Cardinals - Have the prospects and potentially the money.
Pirates - Don't have the money to extend.
Reds - Don't have the prospects or the money.
Cubs - Have money and prospects.
Brewers - Don't money or prospects.
Dodgers - Don't have the prospects, but if they eat enough money on someone like Kemp...
Rockies - Don't have the money or the prospects.
Padres - Don't have the money or the prospects.
Giants - Don't have the prospects.
Diamondbacks - Have the prospects, money is questionable.
So that leaves the Mariners, Rangers, Red Sox, Nationals, Mets, Cardinals and Cubs with the White Sox, Dodgers and D-Backs as potential suitors. The Mariners only really have pitching prospects, and are probably better off improving the offense rather than just adding another big name pitcher. The Rangers have the right prospects but are in need of much better offense rather than more pitching. The Red Sox are pretty solid all the way around and have rookie arms about to break in and will Tampa deal within the division? The Mets could give up future guys like Synderrgard (sp?) to get Price, which would be formidable going forward with Harvey and Wheeler, but I bet Wheeler is dealt to get him. Mets also need some offense at 1B and a much better OF. Cardinals have the right pieces and the money, but are in the Red Sox position with great arms coming up. It's already been posted that offering big extensions isn't really the Cards MO. The Cubs have all the right pieces, and could trade Rizzo to get Price. Might not be Theo's style to trade prospects to get a big name pitcher though, especially with the team still growing. The White Sox would have to trade Sale to get Price, and honestly that doesn't make much sense. The Dodgers have too many outfielders, and they probably are going to hold onto Puig. Ethier, Crawford and Kemp are all very expensive and tons of money would need to be eaten by the Dodgers just to hold court with the Rays to obtain Price. The Dodgers farm is becoming a bit barren as well. The D-Backs could trade young pitchers like Archie Bradley and Tyler Skaggs to get Price, but the extension is always a concern. The Nats, last but not least, might be able to come up with a creative package but it likely will empty what ever farm system they re-built after the Gio trade.
I'd say the "players" on Price turn out to be the Rangers, Mets, Cubs and Nats. I wouldn't be surprised to see price in Queens next year.