The world number one struggled with the injury during his run to the Wimbledon quarter-finals last month and is currently taking time away from the game in order to rest.
Murray's withdrawal means that Rafael Nadal and Roger Federer will be the top two seeds in Montreal, with the former having the chance to move back top of the world rankings if he reaches the semi-finals.
"Sadly I've had to withdraw from Montreal, it's a tournament that I have a lot of great memories from. I'm working as hard as I can to get back on the court as soon as possible," Murray wrote on his official Facebook page.
Murray has won the Rogers Cup three times in the past.