A spokeswoman for Residential Management Ltd said the company behind the facility put it into voluntary isolation for the rest of the weekend as part of "standard procedure".
It had also taken precautionary Covid-19 tests.
"We had some residents today who were experiencing flu-like symptoms," she said.
"One resident had a mild temperature this morning but that has since returned to normal."
"Regardless, those residents are in isolation and we've let the families of our residents know that we aren't permitting visitors this weekend, or until we receive the results of the precautionary Covid-19 tests that have taken place this afternoon."
She said the facility's management team was also in regular contact with its GP and the Ministry of Health.
"We'll be keeping a close eye on our other residents in case they develop any symptoms," she said.
"The residents who have exhibited symptoms today are comfortable and not in need of hospital treatment."
There are currently no known community cases of Covid-19 in New Zealand.
An Air New Zealand crew member was a new positive case in the country on Saturday after arriving in New Zealand on December 9, on a flight from the US.
The person tested positive as part of compulsory testing required within 48 hours of operating to the US.
The risk to the general public is considered low, the Ministry of Health said in a statement.
The Covid-19 positive crew member has been self-isolating in their room in a facility used by Air New Zealand for air crew and is being transferred to the Auckland quarantine facility.