Bourque was selected by the Nashville Predators in the fifth round (No. 132) of the 2009 NHL Draft and made his professional debut with Milwaukee of the American Hockey League in 2010 after three seasons with Baie-Comeau in the Quebec Major Junior Hockey League.
He made his NHL debut with the Predators on Dec. 28, 2011, and scored his first NHL goal against the Dallas Stars on Jan. 5, 2012. However, he was limited to 22 games in 2015-16 because of illness and injury and became a free agent. He agreed to a professional tryout with the Colorado Avalanche on Sept. 1, 2016, and signed a one-year contract on Oct. 10, 2016.