There have been no community cases of Covid-19 in the Cook Islands, which has a double-dosed vaccination rate of 96 per cent of the eligible population.
Seventy per cent of those eligible have also received their booster.
A spokeswoman from the Cook Islands' Ministry of Health told Newstalk ZB she wasn't aware the flight was a location of interest, so didn't have any details on whether the infection originated in the island chain or in New Zealand.
She was trying to find out more late on Saturday night.
The Cook Islands only opened to quarantine-free travel with New Zealand days before transmission of the highly infectious variant was detected in the New Zealand community last month.
The country's government decided to keep the existing quarantine-free border with New Zealand open, albeit with extra Covid-19 prevention measures in place.
Travellers must have received their second vaccination at least 14 days before departure and have evidence of a negative Covid-19 test before departure to be eligible for quarantine-free travel.