The Mariners are 51-44, and in pole position for the second AL wild card spot. Last offseason saw Seattle spend big on second baseman Robinson Cano, who signed a 10-year, $240 million contract with the team, and closer Fernando Rodney, who signed for two years and $14 million.
Trade rumors have swirled around Price since long-term extension talks stalled between the left-hander and the Rays last offseason. Price is eligible to become a free agent in November 2015.
Price made his fourth career All-Star game this year, after going 9-7 with a 3.23 ERA and 10.0 K/9 rate in the first half. Zobrist, a two-time All-Star, is batting .266/.352/.401 with six home runs this season.
Morosi also reported that the Mariners began talks with the Rays after their attempts to acquire Phillies outfielder Marlon Byrd stalled.
The MLB non-waiver trade deadline is July 31.
- Alex Hampl