The Scot suffered the problem in Tuesday's 3-0 win over AFC Wimbledon and had to limp off in the second half.
He underwent a scan at the end of last week and boss Paul Warne does not expect him to be back before December, though he might not miss too many league games.
Warne said: “It's a bad result. I think he will be out for six weeks.
“An innocuous landing and the force through his leg made his hammy tear quite significantly.
“They are saying six weeks but that will only be about six league games because of the international break and the FA Cup game, I don't think he will miss a lot.
“We won't see him in a Rotherham shirt for six weeks.”
Lindsay, who suffered a similar injury last season, is a player who becomes especially frustrated when he is injured.
He travelled with the squad to Cheltenham for Saturday's 2-0 win and Warne believes his fighting spirit could see him make a speedier return.
“He travelled down this weekend, he didn't want to miss out, trained this morning in the pool and did gym work,” Warne said.
“He doesn't want to be not involved.
“When I saw him Friday he seemed a bit better, but life is too short to be unhappy for too long.
“He doesn't take injury well, he wants to play every minute. He's a competitor. If the physio says six to eight weeks, knowing the man he is it will definitely be six not eight.
“Hopefully he comes through no problems and look forward to seeing him back.”
The Millers are back in action in the Papa John's Trophy against Scunthorpe on Tuesday.