Three of the 10 cases are mariners on board the Viking Bay ship which is quarantined at Queens Wharf in Wellington.
"Welfare and health checks are being carried out daily, and plans are in place should any of them become unwell," said the ministry in its 1pm update.
There were a further two cases to report which are deemed historical and not considered infectious. A further case from India reported on Friday as historical has now been reclassified as "under investigation" and has been removed from New Zealand's tally at this stage.
Fourteen previously reported cases have now recovered.
The number of active cases in New Zealand is 45 and the total number of confirmed cases since the pandemic started is 2458.
Playa Zahara update
The fishing ship Playa Zahara arrived into Lyttelton Port this morning and is now secured at a quarantine berth within the port exclusion zone, says the ministry.
"This means it is in an area that is not accessible by the public. A comprehensive safety plan has been developed and the risk to port workers and the public is deemed low."
Testing carried out at Port Taranaki on July 13 found 16 of the 18 crew members positive for Covid. All crew members will be retested today.
"In accordance with the plan developed by Canterbury DHB's public health team and the Government cross agency team, five of the crew members will isolate aboard the fishing vessel and the remainder will be transported to managed isolation and quarantine facilities in Christchurch."
Welfare and health checks would be conducted on the crew remaining aboard the vessel and plans were in place should any of them become unwell. Customs would maintain a presence at the port throughout the operation, said the ministry.
Victoria update
The ministry said New Zealand public health officials were closely monitoring the situation in Victoria where there has been a growing number of Covid cases.
Anyone who was in Victoria since 8 July is encouraged to keep checking the Victoria Health website (https://www.coronavirus.vic.gov.au/exposure-sites) for locations of interest which continue to be announced.
People who have been at these locations at the relevant time should immediately isolate at their home or accommodation and call Healthline on 0800 358 5453 for advice on testing.
