Henderson picked up a thigh injury in the 2-0 win over Leicester City last weekend.
The on-loan Manchester United goalkeeper has played 20 times in the Premier League this season to help Steve Cooper's side to 13th on the standings, five points above the relegation zone.
Under the terms of his loan agreement, Henderson was already set to miss two matches against parent club Manchester United in the Carabao Cup semi finals but will now also sit out upcoming games against Bournemouth, Leeds United, Fulham and Manchester City in addition.
"We have two League Cup games [against United] that Dean wouldn't have played in anyway because of the loan ruling so we were always planning without him for those games," Cooper said.
"We need to get this weekend out of the way and see where we're at after that. Dean is going to be out for four to five weeks, which is obviously a blow for all of us, especially him, but these things happen.
"We carry on with our work and will support Dean as best we can to get fit as quickly as possible.
"In the meantime, we have to concentrate on getting ready for games."
Henderson could return to the side against West Ham United in late February.
