With Jaime Garcia recently electing for season-ending surgery and with fellow starters Michael Wacha and Joe Kelly having been hit by injuries this season, the Cardinals could be looking to bolster their rotation depth.
Since the Red Sox are currently 39-50 and in last place in the American League East, it's possible Boston might be looking to become a seller and unload several assets.
Peavy, who came to the Red Sox in a three-team trade from the Chicago White Sox at the deadline last season, has struggled this season, going 1-7 with a 4.64 ERA. But the 33-year-old has shown improvement in his last two starts, allowing just two runs in each, and he has experience pitching in the National League, having spent his first seven-plus seasons with the San Diego Padres. That could make Peavy an attractive trade option for the Cardinals.
Only signed through 2014, Peavy wasn't expected to return to Boston next season.
The Cardinals entered Tuesday at 48-42, four games behind the Milwaukee Brewers in the National League Central.