Valiev put pen to paper on a one-year, two-way deal with Calgary on Tuesday.
Valiev filed for salary arbitration Friday.
Valiev earned a qualifying offer from the Flames on Tuesday.
Valiev was promoted from AHL Stockton on Wednesday as part of the Flames' Black Aces call-ups.
Valiev was demoted to AHL Stockton on Saturday, TSN reports.
The Flames recalled Valiev from AHL Stockton on Saturday, Pat Steinberg of Sportsnet 960 The Fan reports.
Valiev was reassigned to AHL Stockton on Monday.
Calgary recalled Valiev from AHL Stockton on Saturday.
Valiev -- along with Matt Taormina -- was acquired by the Flames from the Canadiens in a swap for Brett Kulak.
Valiev was placed on waivers by the Canadiens on Tuesday, Stu Cowan of the Montreal Gazette reports.