Staunton was passed fit after recovering from a thumb injury, but it was felt that without any recent match practice, he would benefit more from playing games at the Churchill Cup.
Ireland coach Eddie O'Sullivan said: "Jeremy has not played for some time and with no game-time I thought it would be better for him to get some match practice with the 'A' team."
The Ireland squad are in camp in Limerick this week ahead of their summer tour of Argentina, which begins on Saturday, May 26 in Santa Fe.
Ticket information for the two Tests in Argentina is available here.
Tickets are also now on sale for Ireland's World Cup warm-up game against Scotland in August - click here.