The deal is reportedly worth $4.9 million per year, according to Renaud Lavoie of TVA Sports.
Galchenyuk scored 17 goals and added 27 assists in 61 games last season, after posting 30 goals and 26 helpers in 2015-16.
He also missed a stretch of 18 games through December and January this past season after suffering a knee injury. Prior to the ailment, he'd registered 23 points in 25 games.
The Canadiens drafted Galchenyuk third overall in 2012.