The NHL Broadcasters' Association voted for the league's top coach after the regular season. The winner will be announced June 24 in Las Vegas.
Babcock led the Red Wings to a 93-point season and their 23rd consecutive playoff berth, the longest active streak in North American pro sports.
Cooper is in his first full season as a head coach. He guided the Lightning to second place in the Atlantic Division a year after they were 28th in the overall standings.
Roy, in his rookie season as an NHL coach, helped Colorado finish third in the overall standings after placing 29th the year before.