Summer transfer interest in Sunderland striker
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Grigg was due to see a specialist earlier this week to see if surgery would be needed on the grade-three tear.
The Rotherham Advertiser report Warne is planning to try to sign him on a permanent deal in the summer, with Rotherham looking set for a return to the Championship as they lead League One.
Speaking at the weekend, Warne said: “The best-case scenario is eight weeks and it could be worse.
"They're the cards we've been dealt. He may need an op.
"He's limping, which is obviously a bad sign.
"Sometimes you have a muscle injury where your walking is fine and it's only running at speed that bothers you. It shows the enormity of the injury."
Warned added: “He's an experienced pro and knows injuries happen.
"He's disappointed because he's played a big part in our campaign.
“We're now getting to the last stretch where we're going to need our best players, so his absence is a massive blow.
“He's good in the dressing room. He's a calming influence and good around the younger players.”
The 30-year-old striker had started four successive League One games for Rotherham but suffered the setback during last week’s 1-0 win at AFC Wimbledon.
Grigg, who joined Sunderland in 2019, will be out of contract at the Stadium of Light this summer and is highly unlikely to be handed another chance on Wearside.
The frontman has scored two goals in 19 League One appearances for Rotherham this season.
Sunderland, meanwhile, have a battle on to secure promotion this season.
They dropped more points at the weekend, drawing 1-1 with AFC Wimbledon, though it did at least stop the run of three straight defeats.
Next up is the visit of MK Dons to the Stadium of Light on Saturday afternoon, Alex Neil’s first home game in charge since replacing Lee Johnson.