Henderson, 28, was carried off on a stretcher during the 5-1 Championship loss to Norwich and underwent surgery on Monday.
"It's a big loss to the club to lose someone of Stephen's stature," Forest boss Gary Brazil told BBC Nottingham.
"We know how well he'd been playing for us."
Jordan Smith replaced Henderson at Carrow Road, although Vladimir Stojkovic and Dimitar Evitimov are options to play in goal in Tuesday's Championship game at Fulham.
Henderson suffered the injury as he tried in vain to keep out Norwich's third goal, which came in the 18th minute.
Brazil added: "It's disappointing for the boy, to know that's his season over now. The rehab will start almost immediately. He's in really good hands.
"We know what a good goalkeeper he is but we have to move on. It gives someone else an opportunity and hopefully they will grab it."
Nottingham Forest goalkeeper Stephen Henderson will miss the rest of the season with the Achilles injury he suffered on Saturday.