In a video posted on Facebook, Prime Minister Mark Brown said the person was infectious while on the island, and it's" likely that the virus is in our community".
He warned that there would be 'silent transmission' among citizens due to the country's high vaccination rate.
The Pacific archipelago has never recorded a community transmission case of COVID-19.
Prime Minister Brown said the traveller had spent eight days on the Cook Island had tested positive for the Omicron variant when returning to New Zealand.
According to Cook Islands data, 99.6% of the population aged 12 and over are double vaccinated. 70% of those eligible for a booster have received their third shot.