The case's partner has returned a negative test, and three friends who were also travelling with them are yet to be tested.
"It is likely that the person who tested positive was infectious while here and further likely that the virus is in our community," a statement released by Cook Islands Prime Minister Mark Brown said.
"It may be there is 'silent transmission', where our high vaccination rate is so protective that people get Covid but so mildly that they do not realise they have it."
The infected person spent eight days in Rarotonga and first began developing symptoms in the early hours of Monday, February 7, the same day they departed from Rarotonga to go back to New Zealand.
The case is fully vaccinated.
According to the Cook Islands Prime Minister, the country's high vaccination rate will give the population "substantial protection from serious illness".
What is known of the case's movements in Rarotonga
The statement detailed what the Cook Islands government knows about the positive case's movement in Rarotonga.
On Friday, February 4 , they visited Trader Jacks at 11.30pm and at 12.42am on the Saturday morning (February) they tagged into the Rehab nightclub.
"We do not need everyone who was at either of those places around those times to come forward for testing," the Prime Minister stated.
"But if you have developed flu-like symptoms, like headaches, a sore/scratchy throat, fever, or body aches, you should come to the drive through testing station at Tupapa Public Health Building today and get tested."
The Cook Islands border will remain open with the current protocols in place.
"Te Marae Ora may put in place additional Public Health protocols and if they do, these will be communicated to you," the statement read.
"You can also expect existing regulations and protocols to be actively enforced over the coming days.
"Te Marae Ora has begun the contact tracing process and testing of close contacts, that is, people who were working at venues visited by the case and who are considered to be close contacts.
"Most of all we need to stay calm, wear masks, maintain physical distancing and good hygiene practices."
