National selection manager Kim Littlejohn said the 32-year-old, who claimed two five-wicket hauls in New Zealand's most recent tests against South Africa, suffered bone bruising and ligament damage after twisting his ankle while running.
Littlejohn said no decision has yet been made on who will travel to the West Indies as Gillespie's replacement.
New Zealand will play the first of five one-day internationals against the West Indies on Friday.