The Canadiens have had contact with the pending unrestricted free agent, who would be very interested in playing for them, according to Sportsnet's Eric Engels.
Montreal only has about $8.7 million in cap space, but they also only have four pending free agents. Perron will become a UFA if the Anaheim Ducks don't re-sign him before July 1.
Hagelin helped the Penguins win the Stanley Cup earlier this month, forming the productive "HBK" line with Nick Bonino and Phil Kessel, while Perron managed only 16 points in 43 games with Pittsburgh before collecting 20 points in 28 regular-season contests with the Ducks down the stretch.
Perron scored a goal and added two assists in seven playoff games.
The 28-year-old is coming off the final season of a four-year, $15.25-million deal he signed with the St. Louis Blues in 2012.