Bradley was placed on the 10-day disabled list Wednesday. He was injured while reaching for home plate during a slide in Tuesday night's game against the Cleveland Indians.
Bradley was sent back to Boston for an MRI on Wednesday. Manager John Farrell said the tests showed no tears or fractures.
Bradley won't be ready to come off the disabled list in 10 days, but the Red Sox are hopeful he will return before the postseason.
"It's hard to tell at this point," Farrell told reporters when asked if there's a general timeframe for Bradley's return. "I wish I had more of a definitive answer, but I just don't know."
Farrell said he's concerned whenever a hitter is dealing with a hand injury.
Bradley slid around catcher Yan Gomes' tag to score on Eduardo Nunez's seventh-inning double. Indians manager Terry Francona challenged the call, which was upheld after a review. Farrell said had the call been overturned he would have challenged that Gomes blocked the plate and didn't give Bradley a clear lane to slide.
"It's unfortunate that he wound up in an unfortunate position on the slide," Farrell said.
Andrew Benintendi was starting in center field with Brock Holt in left Wednesday. Chris Young will also see action in left field, and Farrell said Nunez may see time there as well when second baseman Dustin Pedroia (left knee inflammation) returns from the DL.
Pedroia remained in Boston while the team went on road. He's been running on the treadmill and hitting in the cages, but he hasn't begun any on-field work.
To take Bradley's spot on the roster, the Red Sox called up infielder Deven Marrero from Triple-A Pawtucket.
Bradley was batting .262 with 14 home runs and 54 RBIs this season, hitting .400 in his past seven games.
Marrero has played 58 games in three stints in the majors this season. He has batted .212 with 3 homers and 23 RBIs with the Red Sox.
ESPN's Scott Lauber contributed to this report.