The award is presented to the head coach who has contributed the most to his team's success.
Babcock led the Red Wings to a 93-point season and their 23rd consecutive playoff berth, the longest active streak in North American professional sports.
Detroit overcame a franchise-record 421 man games lost due to injury. The Red Wings had nine players make their NHL debuts this past season and used 38 different players in all.
Cooper, in his first full season as head coach, helped the Lightning record 101 points. Tampa Bay finished second in the Atlantic Division a year after placing 28th in the overall standings.
The Lightning were 20-11-9 in one-goal games and posted 21 road wins, one shy of the franchise record.
Roy lifted the Avalanche to a historic turnaround in his rookie season as an NHL head coach, helping the team finish third in the overall standings with 112 points after placing 29th overall the year before.
Colorado became the first club since the NHL expanded to 21 teams in 1979 to go from the bottom three to top three in a single season. The Avalanche matched a franchise record for wins, had the league's best road mark and did not suffer a regulation loss when leading after two periods.
Members of the NHL Broadcasters' Association submitted ballots for the award at the conclusion of the regular season. The top three vote-getters were designated as finalists.
The winner will be announced June 24 at the NHL Awards in Las Vegas.