Kaino was omitted from a 31-man squad leaving for Argentina Friday for the October 1 Test in Buenos Aires.
Waisake Naholo, who pulled a hamstring in the opening Championship match against Australia, and George Moala, who injured a knee during New Zealand's June series against Wales, were named in the squad.
Kaino picked up his injury when he fell awkwardly in a tackle during Saturday's 41-13 victory over South Africa in Christchurch, which delivered New Zealand the Championship title.
Team management hope he will be fit for the return leg against the Springboks in Durban on October 8.