Rychel, born in Torrance, Calif., and raised in Tecumseh, Ont., was drafted by Columbus in the first round (19th overall) in the 2013 NHL Draft.
The 23-year-old had 18 goals and 24 assists last season in 71 American Hockey League games split between Toronto Marlies and the Laval Rocket. He had a goal and an assist in four games with the Canadiens.
Rychel has three goals and 11 assists in 41 career NHL games with Columbus and Montreal.
The Flames acquired him in a trade with Montreal for winger Hunter Shinkaruk earlier on Monday.