Alzner has two years remaining on his contract with a cap hit of $4.265 million.
The veteran rearguard also skated in 87 games with the AHL's Laval Rocket, posting two goals, 19 points and a plus-6 differential.
Alzner joined the Canadiens as a free agent in 2017 after nine seasons with the Washington Capitals.
Also Tuesday, the Detroit Red Wings announced that forward Justin Abdelkader has been placed on waivers for the purpose of buying out his contract. The 33-year-old recorded three assists during the 2019-20 campaign. Abdelkader was drafted by the Red Wings in the second-round of the 2005 NHL Draft, and has since spent 13 seasons with Detroit.