Grigg, who joined Sunderland in 2019, will be out of contract at the Stadium of Light this summer and is unlikely to be handed another chance on Wearside.
The frontman has scored two goals in 19 League One appearances for Rotherham this season.
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"Griggy coming off injured is a big blow. I think he is the glue to the team really," said Millers boss Paul Warne ahead of Sunday’s trip to Sheffield Wednesday.
"His link-up play is really good and he works well with Smudge (striker Michael Smith).
"Unfortunately, he's going to be out for a long period of time, so that isn't good news.
Warne continued: "I don't really like to put a timeframe on Griggy's because the physios told me, but we think he might be about eight weeks.
"His is a severe hammy. There was obviously nobody near him when he did it. He ran and put the cross in for a chance that Smudge should definitely have taken.
"It's a massive blow. Griggy has got back to looking really sharp and his offside goal was very harsh as well.
"He'll be out for a period. As I always say, it's 'next man up' but I do say that with a heavy heart because I do think we're going to miss him."