The 30-year-old had his ironman streak of 622 games snapped earlier this season when he was a healthy scratch, a lineup designation the Barnaby, B.C, native has had to bear for much of this year.
Alzner did not skate with the team at practice on Monday morning in Brossard, Quebec, with captain Shea Weber joining regulars David Schlemko, Brett Kulak, Jeff Petry, Victor Mete, Jordie Benn, Xavier Ouellet and Mike Reilly on the ice.
The Canadiens need to clear to a spot on the 23-man roster ahead of Webers return, which is expected to come Tuesday night against the Carolina Hurricanes. Weber was paired with Schlemko during Mondays practice.
The 30-year-old Alzner signed a five-year, $23 million contract with the Canadiens as an unrestricted free agent in 2017 after nine seasons with the Washington Capitals.
He has one assist in eight games with the Canadiens this season and 12 points in 82 games last season.
Alzners $4.625 million cap hit is the third-highest among Canadiens defencemen, behind Weber and Petry. Its the sixth-highest on the team.
In 681 career NHL games with Washington and Montreal, Alzner has 20 goals, 130 points and a plus-56 rating.
Alzner has this year, and three more years left on his deal, which also includes a seven team no trade list. If he clears waivers for the AHL, the team saves $1,025,000 on the cap as his cap hit will then become $3,600,000 while in the minors.