Bruce confirmed the news in a press conference on Thursday, ahead of Hull's Premier League clash with Crystal Palace at the KC Stadium on Saturday.
"We're just delighted he doesn't need an operation, because we thought it might," he said.
"The specialist has said he needs to rest, and stop diving on it. Hopefully, in four to six weeks he'll be back to normal."
McGregor's absence opens the door for 39-year-old Steve Harper to step into the breach.
The former Newcastle United man made 19 appearances in all competitions last season, but has featured just once during 2014-15, in the 3-2 defeat at West Brom in the League Cup third round last week.
"Harper will give us that experience that we saw last year, he came into the fray and did fantastically for us," Bruce added.
"I wish he was a few years younger, I'm sure he does too but he's a very good goalkeeper."
