The 32-year old Scotland international has been battling a shoulder problem for a few weeks now and has been told by medical specialists that he needs rest in order to cure the issue.
McGregor, who joined the Tigers from Besiktas in July 2013, will miss the next four weeks of his side's Premier League season as well as Scotland's Euro2016 qualifiers with Georgia and Poland later in October.
"Allan is going to be sidelined for four weeks and, of course, it's a big blow to lose him," Bruce said ahead of this weekend's meeting with Crystal Palace.
"He's been playing under a cloud and even though he's as tough as they come, it's got to the stage where we have to take him out of the firing line because that's the only way to make sure he recovers.
"The positive thing is that he doesn't require an operation to solve the problem.
"We feared at first that he would need surgery, but scans have confirmed that rest is the cure."
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