Hickey didn't receive a qualifying offer from the Sabres and will become a UFA on Friday, Sabres beat writer Joe Yerdon reports.
Hickey was demoted to AHL Rochester on Monday.
Hickey signed a two-year, entry-level deal with the Sabres.
The Coyotes traded Hickey along with Mike Sislo to the Sabres on Thursday in exchange for Hudson Fasching.
The Flames traded Hickey, goalie Chad Johnson, and a conditional third-round pick to the Coyotes on Saturday, with veteran netminder Mike Smith heading the other way, TSN reports.