According to Michael Silverman of the Boston Herald, Betts is considered day-to-day.
The star outfielder exited in the fifth inning and was replaced by Rajai Davis.
Betts has been fairly durable during his young MLB career, but this marks the second time he has dealt with an injury this season after hurting his knee during an August game against the Cleveland Indians.
Although Betts' production is down this season, he is still having a strong year with a .263 batting average, 22 home runs and 90 RBI.
Betts developed into an elite player last season when he hit .318 with 31 home runs, 113 RBI, 122 runs and 26 stolen bases, all of which were career highs.
At the conclusion of the campaign, Betts won the Silver Slugger Award and a Gold Glove, and he finished second in American League MVP voting behind Los Angeles Angels outfielder Mike Trout.
Betts is one of the few five-tool players in Major League Baseball, and his importance to the Red Sox is immeasurable due to his ability to contribute in all offensive and defensive facets.
This is a crucial part of the season, and the Red Sox can ill afford to be without Betts for an extended period of time.
If Betts does land on the shelf, Andrew Benintendi will have an even greater responsibility to produce in the outfield, while Davis and Chris Young are likely to see expanded roles with utilityman Brock Holt also getting some opportunities.
Although Boston has depth in the outfield, none of the potential replacement options boast a skill set that comes close to rivaling Betts.
Even if Betts' injury isn't serious, Boston has a three-game cushion over the New York Yankees in the American League, so it can afford to play it safe with him.