Genomic testing results are yet to be confirmed but New Zealand authorities are working on the assumption it is the Delta variant of the virus.
After the weekend visit of an Australian tourist, the Wellington region is now in alert level 2 until at least Sunday evening.
Covid-19 Response Minister Chris Hipkins said Australian authorities were considering the tourist's case to be part of an existing cluster in Sydney.
"They do believe there is a link there. And the genome sequencing would hopefully support that conclusion."
He told Morning Report the case was likely to be the Delta variant of the virus.
Hipkins said the person visited many locations where there were a lot of other people and "that increases the risk [of an outbreak]".
It would be a "successful outcome" if in five days, there were no more cases identified apart from a handful of close contacts.
"The last thing we want to do is wake up on Monday morning and find out that we did not act soon enough.
"If we started to see unlinked community transmission - that's a real trigger point for us."
He said the four close contacts in New Zealand had tested negative.
But test results of the person's partner, who was also now in Australia, are yet to come in.
The testing centre in Taranaki St is fully booked for today, and another one was set up in Hataitai yesterday afternoon.
Hipkins said more sites would be added to try to get everyone who fits the criteria tested today.
"I'm comfortable with the level of vaccination across the country based on the vaccine supply that we've got available.
"We've used up or getting very close to having used up all of the vaccines that we've got available to us."
He said more vaccines were expected in the next few weeks.
National Party Covid-19 spokesperson Chris Bishop said the government was right to move Wellington to alert level 2.
He told Morning Report it was a tricky situation because it was the Delta variant of the virus that was spreading in Sydney.
"The chance of spread is there. This is the cautious thing that the government has done and so we support it."
However, he said the initial response could have been better and more could have been done the night before so fewer people would leave home for work.
As for the vaccine roll-out, he said the government needed to have a contingency plan in place.
"New South Wales and New Zealand have been good at doing contact tracing. The next 24 to 48 hours for New Zealand and Wellington in particular is going to be critical.
"So go and get a test if you have to, follow the public health advice, and of course we've got to keep scanning in."