McGregor has played a key role in Hull's impressive return to the top flight but is now set for a spell on the sidelines after sustaining the problem at Everton on Saturday.
"We've just found out this morning that Allan is going to be out for five or six weeks," Bruce told the club's official website.
"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."
Steve Harper, brought in over the summer after his contract at Newcastle expired, will deputise, starting in this weekend's trip to Tottenham.
Bruce said of the 38-year-old: "This is the reason we brought Steve in to be fair.
"He's got an abundance of Premier League experience and we're lucky that we can call upon somebody with his experience."
Hull go into the clash at White Hart Lane in ninth place, level on points with defending champions Manchester United.