Betts was out of the lineup for a third straight game Tuesday against the Giants in Boston. He said he's hopeful to return this season.
Manager Alex Cora called Betts day-to-day, adding that if he plays this weekend against the Rays on the artificial turf in St. Petersburg, Florida, it would be as a designated hitter.
The reigning AL MVP is hitting .293 with 28 home runs, 78 RBIs and 131 runs scored this season for Boston.
Also Tuesday, Cora said the team is meeting to discuss the next steps for David Price (wrist), with one option being shutting the left-hander down. Chris Sale, meanwhile, is in Florida as he recovers from a left elbow injury and will meet the team in St. Pete during the Rays series that begins Friday.