In the second half of Sale's 22ā20 Premiership victory over Saracens at the Salford Community Stadium on Friday, Tuilagi sustained a grade three tear.
The 32-year-old, who has 60 caps for England, had a persistent groin issue early in his career.
Tuilagi is probably going to miss England's first Six Nations game on February 3 in Rome against Italy.
The next week, they will play Wales at Twickenham.
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