Devan Dubnyk of the Minnesota Wild and Pekka Rinne of the Nashville Predators are also finalists for the award, which is given "to the goalkeeper adjudged to be the best at his position."
The winner will be announced June 24 at the NHL awards ceremony in Las Vegas.
Price led the league in 44 wins while posting a 1.96 goals-against average and .933 save percentage.
It is the first time a goaltender was at the top of all three categories since Ed Belfour of the Chicago Blackhawks in 1990-91.
Price also broke a franchise record for wins in one season set by Jacques Plante in 1955-56.
Ken Dryden also reached the mark in 1975-76. His save percentage was the third-highest in a single season since 1976-77 when the NHL started to track the stat. Price also set a career high with nine shutouts, which ranked second in the NHL this season.
Rinne finished with a 41-17-6 record in 64 appearances for the Predators this season. He tied for second in the NHL in wins, ranked third with a 2.18 goals-against average and seventh in save percentage at .923. It is the third time that he has been a finalist for the award.
Since a Jan. 14 trade, Dubnyk is one of the main reasons why Minnesota is in the playoffs. The Wild sat eight points out of the post-season when they acquired Dubnyk and have been solid ever since.
He set a franchise record for 38 straight starts and went 27-9-2 in that span. Dubnyk also had a 1.78 goals-against average, .936 save percentage and five shutouts since joining Minnesota. It is the first time that he is a finalist for the award.
The Vezina trophy is in memory of Georges Vezina, a former Canadiens goaltender that collapsed during an NHL game on Nov. 28, 1925. He died of tuberculosis a few months later.