Staal, 36, has three goals and 10 points in 32 games this year with the last-place Sabres. Buffalo has lost 16 consecutive games, the longest streak of the shootout era (i.e. since 2005-06).
Sportsnet's Chris Johnston noted that Staal is "believed" to be subjected only to Canada's reduced seven-day quarantine period, though there hasn't been confirmation from the government on that yet.
This year, Staal carries a $3.25 million cap hit. He'll be an unrestricted free agent at season's end.
The Thunder Bay, Ont., native is in his 17th NHL season. He has a quartet of 20-goal campaigns in his 30s, highlighted by a 42-goal season with the Minnesota Wild in 2017-18.
The Canadiens are his fifth franchise, following the Carolina Hurricanes, New York Rangers, Wild and Sabres.
Staal has 51 points in 62 career playoff games, and he led the Hurricanes to their lone Stanley Cup victory back in 2006.