The ship's crew members will be put into managed isolation when they return to land.
One of two mariners who tested positive for Covid has the highly infectious Delta variant, health officials have confirmed.
The first whole genome sequencing result for the two infected mariners confirms that one of them has the Delta variant.
This sequencing also shows this infection is not linked to any other cases in New Zealand.
The Ministry of Health said an "all of Government" response had been set up, led by Customs.
Health was working closely with the dedicated response, border agencies, port authorities and the shipping company to address the health needs of crew members and to determine the next steps for the vessel.
The mariners are on a ship off the coast of Taranaki.
It was not yet clear which port the ship will return to.
On their return to New Zealand, 15 of the 20 crew members disembarking, including the two infected mariners, are expected to go into a managed isolation and quarantine facility.
The Ministry of Health said this will likely be a facility at the port they return to.
There were no managed isolation or quarantine facilities in New Plymouth so at this stage it was considered unlikely they will return to Port Taranaki. New Zealand Customs and other agencies were now working on the operational details of the boat's return.
Health has also been advised that a second foreign-owned fishing vessel operating from New Zealand has contacted authorities and declared that a number of crew have reported flu-like symptoms and the vessel has requested to dock in New Zealand.
The vessel was outside our territorial waters and presented no immediate risk to New Zealand. This situation would be updated by Customs as more information came to hand, health officials said.
One new case in MIQ today
Meanwhile, there is one new Covid case in managed isolation to report today.
The latest person to arrive in New Zealand and test positive for Covid came from Russia on July 6. They tested positive on arrival and were in a managed isolation facility in Auckland.
There are now 41 active cases in New Zealand.
The Ministry of Health has confirmed the pause on quarantine-free travel from Queensland and New South Wales to New Zealand will continue.
It said the decision followed a risk assessment by public health officials which determined there was still a need to get a better understanding of the developing situation and the number, and pattern, of cases being reported in these Australian states.
A range of factors for each state were considered, including whether any new cases are identified, whether they were infectious in the community, and the results from Covid-19 testing of any contacts and from the wider community.
New Zealand public health staff remained in close contact with Australian health authorities about the actions being taken in each state. As with previous pauses, it would be under constant review.
"We consider this pause to be prudent while investigations continue and until further test results are returned," said the ministry.
Meanwhile, the Government has also announced plans for the carefully managed commencement of return green flights from Queensland and New South Wales to New Zealand, from Friday at 11.59pm.
These were open to new Zealand citizens and select visa holders and residents. Anyone boarding these flights would need to have a negative pre-departure test taken within 72 hours before departure.
Health officials continued to regard the situation with the two infected sailors on board the Spanish-flagged ship as low risk to public health due to the infection prevention control process in place. There remained no associated locations of interest.
Today's update comes after three cases in managed isolation reported yesterday. There remain no cases of Covid in the community anywhere in New Zealand.
All three new cases reported yesterday are carrying out their 14-day quarantine period in facilities in Auckland.
One of the returnees arrived on a flight from the United Kingdom, via the United Arab Emirates, on June 23. They returned a positive result for the virus on day 12 of their routine testing.
The second recent arrival touched down in Auckland from Malaysia, via Singapore, on June 26. They tested positive on day eight of their stay and is a contact of a previously reported case.
The third person arrived from the UAE on a direct flight last Friday (July 2). They returned a positive result on day three of their regular tests.
Two crew members on a fishing vessel currently just off the coast of Taranaki tested positive for Covid earlier this week and now they and at least 13 other crew remain isolated at sea.
The Maritime Union yesterday confirmed that the vessel involved was the Spanish-flagged Viking Bay.
There is some confusion about what will happen to them next; after Port Taranaki said it had not been contacted by the Ministry of Health about any plans to return the crew to port.
That is despite the Ministry of Health saying the opposite - announcing yesterday that the vessel would be returning to port and all on board would go straight into managed isolation and quarantine.
Port Taranaki refuses fishing vessel due to higher risk
Port Taranaki later released a statement, with chief executive Guy Roper saying: "Throughout the Covid-19 pandemic and our response to it, the health and safety of our staff, other port users and the Taranaki community has been our priority.
"We have followed all the Ministry of Health protocols, including those for crew transfers. However, we believe having the vessel return with known Covid-19 cases aboard is a completely different scenario and puts our staff and the community at a higher risk."
Roper said they also had to consider staff numbers and their customers.
They had a limited number of pilots, he said, and the pilot on duty for the vessel's return would likely have to self-isolate as well.
"There is also no certainty about how long the fishing vessel would need to remain berthed without appropriate manning.
"For those reasons, we have decided that the vessel will not be returning to Port Taranaki."
Three people have been identified as contacts of the two mariners - a port pilot, a customs officer and the bus driver who drove the crew of nine mariners from Auckland to New Plymouth.
All three contacts were said to have been wearing full personal protective equipment when they interacted with the mariners, the MoH said, while the bus driver wore a N95 mask for the entire journey.
The bus driver has since been put into an Auckland quarantine facility, while the port pilot and customs office are self-isolating. All three were due to be tested for Covid yesterday.