Scolari will play Yogi Berra, Major League Baseball player-turned-manager, and Shayne will play his wife, Carmen, in the show, which is to open at Circle in the Square Feb. 6.
Preview performances are to begin Jan. 10.
"Generations of Yankees' fans have been part of the winning tradition since their early childhood. Their emotional commitment and passion will be fully realized as they experience this play and sit with the players who forged that history," producer Tony Ponturo said in a statement.
"We are excited to bring the legacy of the New York Yankees and their proud tradition to the equally proud tradition of the Broadway theater," producer Fran Kirmser said.
The play was written and is being directed by Eric Simonson.
Scolari was last seen on Broadway in "Lucky Guy," opposite his "Bosom Buddies" co-star Tom Hanks.
Shayne returns to Broadway following performances as Roxie Hart in "Chicago," Cosette in "Les Miserables" and multiple roles in "A Chorus Line."