Cozart is in fourth place among shortstops with 431,365 votes. Another Cardinal, Jhonny Peralta, is in first place with 595,741.
Phillips is in fourth place at second base with 337,146 votes behind leader Dee Gordon (934,249) of the Marlins.
Votto rounds out the pack, currently in fifth place at first base with 384,285 votes. Adrian Gonzalez of the Dodgers leads the voting with 929,016.
Cozart is looking for his first All-Star Game appearance, while Frazier was selected to his first career All-Star Game last season. Phillips is a three-time NL All-Star, and Votto is a four-time NL All-Star.