1:01pm: A source tells Ryan Divish of the Seattle Times that any trade for Price would have to include Walker (Twitter link).
11:47am: In the wake of Seattle's historic ten-year agreement with Robinson Cano, one Major League executive told Bill Shaikin of the L.A. Times that he is "convinced" the Mariners' next move will be to make a push for David Price (Twitter link). Jon Paul Morosi of FOX Sports tweets that the Mariners believe they have the prospects to land Price, and indeed, that would seem to be the case.
This is only my speculation, but Seattle could look to build a package around top prospect Taijuan Walker and the recently displaced Nick Franklin, who could be deemed expendable with Cano in the fold for the Mariners. The Rays, of course, will likely be open to moving Price this winter. He projects to earn $13.1MM in arbitration (per MLBTR's Matt Swartz) and is controlled through the 2015 campaign. It would make sense for the Mariners to aggressively pursue short-term upgrades in the early years of Cano's contract in order to maximize their chances while he is still in his prime.
Morosi also notes (on Twitter) that the Mariners are still in pursuit of Shin-Soo Choo, Carlos Beltran, Nelson Cruz and other free agents. Morosi also reported earlier that the Mariners are one of the teams that is currently in on Mike Napoli, who reportedly has an offer in-hand from a club other than the Red Sox that Boston has yet to beat. It's not clear if that club is the Mariners, but their talks with Napoli are said to have been recent. An upgrade at first base, DH or in the outfield still seems likely for Seattle GM Jack Zduriencik, and the addition of Cano could signal to potential free agents that the Mariners are serious about winning.