Blue Jackets GM Scott Howson has already spoken with Mathieu Garon's agent, Pat Brisson, and the two are expected to resume discussions in the coming weeks.
Howson shopped young goalie Steve Mason this year, with teams like Philadelphia knocking on his door, but it's uncertain if a move will be made this off-season.
The Dispatch claims the Blue Jackets prefer to keep Mason, and are holding out hope that he returns to rookie form, when he was named Rookie of the Year in 2008-09.
Columbus already made one move related to its goaltender, as goalie coach Dave Rook will not be returning next season.
The Blue Jackets could also look to deal forward Jakub Voracek around the NHL Entry Draft as part of a package for a proven top-tier player.
Like Mason, Voracek was dangled as trade bait this season.
Meanwhile, the Edmonton Journal reports the will likely let defenseman Jan Hejda walk as an unrestricted free agent this July.