About 250 people who have already left managed isolation have been asked to isolate and get tested because of possible transmission within their hotel.
In a statement, the ministry said six new cases had been found in managed isolation and quarantine (MIQ) today, with no new cases in the community.
The government was investigating the possibility of a transmission at the Grand Mercure managed isolation facility in Auckland, the ministry said.
It said results from genome sequencing had shown a link between two returnees at the Grand Mercure. The infection for both returnees was confirmed as B.1.1.7, the variant first identified in the UK.
The returnees were in separate rooms on different floors and arrived on different flights within two days of each other, the ministry said.
Both people had previously been reported in the ministry's case numbers, and one was related to reports of two bus journeys from the facility to an exercise area.
The ministry said 14 returnees who were also on the bus had been classified as close contacts
No other cases had yet been found to be linked to those two, but investigations were continuing, it said.
Some 250 people who had left the MIQ facility since 10 March were being contacted today, and asked to get tested immediately and isolate at home until returning a negative result.
Additional testing and surveillance of staff was being put in place at the facility.
Covid Tracer app update
The ministry said an update to its NZ Covid Tracer app meant about 100,000 New Zealanders with older iPhones could now use the bluetooth function.
Version 4 had been released to extend Bluetooth support to iPhone 5S, 6 and 6+, it said.
People were reminded to scan the QR code posters as often as possible.
One year since lockdown level 4
Today marks one year since the start of alert level 4 lockdown across New Zealand in an effort to combat Covid-19.
Two recoveries brought active cases to 71, with the seven-day average of new cases at five. Total confirmed cases is at 2120.
Labs processed 6209 tests yesterday, for a seven-day rolling average of 4575.
Yesterday, the government made changes to the way it would charge for managed isolation and quarantine (MIQ).
It also announced its confirmed criteria for applications for an early vaccination - including people who need to travel outside NZ for compassionate grounds, or representing New Zealand in an official capacity.
Compassionate grounds include needing to provide critical care for a dependent, visiting an immediate family member who is dying, or accessing critical care not available in New Zealand. Representing New Zealand must meet a criteria of national significance.
The ministry also reported three new cases in managed isolation yesterday.