An official announcement is expected soon.
Treliving, 44, has been the Phoenix Coyotes' assistant GM for the last seven seasons.
Flames president of hockey operations Brian Burke served as interim GM after firing Jay Feaster in December.
Treliving, a native of Penticton, B.C., is also assistant GM for Team Canada at the IIHF world championship. He is also the son of Boston Pizza co-owner Jim Treliving, who is one of the stars of CBC's Dragon's Den.
Brad Treviling previously served as president of the Central Hockey League and president and director of hockey operations for the Western Professional Hockey League, which he founded.
The Flames missed the playoffs after finishing with a 35-40-7 record for 77 points. It was the fifth consecutive year the team missed the post-season, while posting its lowest point total since since 2002-03.
Calgary was 13th in the Western Conference standings and will be picking fourth overall in the upcoming NHL draft scheduled for Philadelphia on June 27-28.