The ministry said testing was being recommended as a precaution only, and the risk of contracting the virus was low.
The Air New Zealand cabin crew member tested positive after arriving in Shanghai on Sunday. They are being re-tested and the case is classified as "under investigation" by the ministry.
Out of an abundance of caution, it was being treated as a confirmed case.
In the two weeks before the positive test, the person's movements had been confined to Auckland and they had contact with relatively few people.
So far, three close contacts of the person had been identified in Auckland, all of whom were in self-isolation. Another two close contacts were under investigation.
The Air NZ staffer also made brief visits to a number of other businesses and shops on November 20 and 21.
Staff at most of these businesses were being advised to isolate at home and get tested if they developed symptoms. Members of the public who were at these locations were not considered to be at risk.
Further locations may be notified as the investigation continues.
Genomic testing would take place once they returned to determine the possible source.
The crew member had mild symptoms and was being retested. The rest of the Air NZ crew did not have symptoms and had tested negative.
The person's flight schedule in the past month included flights to Australia and China, Hipkins said.
"Coming back from a higher risk place like China - they should be doing a 48-hour stand down and a negative test at the end of the 48 hours before they can go back out into the community.
