Yzerman is in his first season with the Lightning, who are in the playoffs for the first time since 2007. The former Detroit Red Wings star made a number of key moves to help the team, including in the January acquisition of goalie Dwayne Roloson from the New York Islanders.
Vancouver compiled the league's best regular-season record and reached the 50-win mark for the first time in team history under Gillis.
Poile has Nashville in the playoffs for the sixth time in seven seasons.
The NHL awards ceremony is June 22 in Las Vegas.