All of the crew were remaining in isolation, and Air NZ was working with Chinese and New Zealand officials to arrange for them to return to New Zealand.
The spokeswoman said the crew member did not work on any flight between their negative test on Wednesday and their flight on Sunday.
The ministry confirmed it was investigating the possible case, but would not comment further.
Prime Minister Jacinda Ardern said this afternoon that contact tracing was underway for the Air NZ crew member as a precautionary approach.
They were being re-tested and a source would be looked into to ensure everyone who needed to be isolated was in a facility, she said.
Two new cases at the border
Meanwhile, the ministry confirmed two new cases of Covid-19 in managed isolation in New Zealand today. There are no new community cases.
Both of the new cases arrived on November 19 and tested positive around day 3 of their isolation.
One arrived from the United Kingdom via the United Arab Emirates and the country of origin for the second case is still being determined.
There are now 52 active cases of Covid-19 in New Zealand.
Officials' border worker bungle
The Ministry of Health also said that it wrongly reported that a new case linked to the Defence Force cluster was a border worker.
The person was actually a close contact of "Case B", a Defence Force staffer.
There were 3274 tests around the country yesterday, bringing the total number of tests since the pandemic began to 1.23 million.
There are six confirmed cases in the Defence Force cluster.
Ministry's message
The ministry reiterated that at Alert Level 1 anyone who became unwell with cold or flu symptoms anywhere in the country should seek a test quickly, and stay at home.
"If you are unwell, sacrifice your plans and protect your family, friends and colleagues by staying at home and getting a Covid-19 test.
"It is the best way to ensure the freedoms we enjoy under Alert Level 1 can continue into the much-anticipated summer break."
The ministry also said the New Zealand contact tracing app would be updated tonight to make it easier to use.
Existing app users would no longer need to be prompted to log back into the app with their password, and new users would not have to provide their email address.
Yesterday's cases
Yesterday nine new cases were confirmed, all in managed isolation, as New Zealanders returned home in large numbers for Christmas.
They had travelled from eight different countries - an indication of how the pandemic continued to ravage parts of the world, the ministry said yesterday.
There are 50 active cases in New Zealand. There have been 1.2 million tests since the pandemic began.
The Defence Force cluster, which stemmed from a staff member who worked in the Jet Park isolation facility, remained at six confirmed cases yesterday.
The sixth member of the cluster was confirmed on Saturday, and was a close contact of the first Defence Force worker to contract the virus.
Two pop-up testing stations continue to operate in central and east Auckland in response to the cluster: one on High St and one at 292A Botany Rd (Spectrum House).
Meanwhile, a new study has detailed how a single traveller infected four others on a flight from Dubai to New Zealand, despite testing negative before they boarded the flight.
The government-funded New Zealand study into the flight raises questions about the safety of international travel during the pandemic, even when the precaution of pre-flight testing is taken.
The infections occurred on a flight from Dubai to New Zealand in September.
In total, seven cases from the flight were identified in managed isolation and scientists believe that four of those cases picked up the virus during the 18-hour flight.
There were 86 passengers onboard the flight.
Those seven passengers came from five different countries before travelling on the flight from Dubai, but genomic testing suggested that four infections took place onboard as they carried genome sequences from Switzerland, the country of origin of the source case.
