The 24-year-old Swiss defenseman will become an unrestricted free agent on Friday at noon.
Chosen by the Canadiens in the third round (No. 73) of the 2007 NHL Draft, Weber scored five goals and 32 points in 115 career games.
After playing 60 games in 2011-12, Weber had trouble getting in the lineup under new Canadiens coach Michel Therrien in 2012-13, playing just six games and notching two assists.
Weber has shown an ability to help on the power play with a heavy, accurate shot from the point, but he had trouble developing his game in his own end over his three full seasons in Montreal.
Also not receiving qualifying offers from the Canadiens are defenseman Joe Stejskal and right wing Ian Schultz. Schultz is best known as the other forward, along with center Lars Eller, acquired by the Canadiens in the trade that sent goaltender Jaroslav Halak to the St. Louis Blues in 2010.