McGregor suffered the injury during the 2-1 defeat at Everton last weekend.
Hull boss Steve Bruce told the club’s website: “We’ve just found out this morning that Allan is going to be out for five or six weeks.
“As I’ve said repeatedly, injuries are the one area where we needed to stay fit and healthy and this one couldn’t have come at a worse time.
“Allan will be a big miss for us because he has played very, very well and we wish him a speedy recovery.”
