The Ports of Auckland has agreed to allow the vessel with two positive Covid-19 mariners onboard access to its quarantine anchorage.
The company said it would not be taking the Spanish flagged fishing vessel Viking Bay "in-port" but would allow it access to its quarantine anchorage near Rangitoto Island.
Ports of Auckland spokesman Matt Ball said if Viking Bay needed to bring people ashore, it would need to be done via another small transfer vessel organised by the ship itself.
"The Ports of Auckland has just received a request for the Viking Bay to come to Auckland," Ball said.
"If the vessel comes to Auckland it would not berth in the port but would go to one of the two quarantine anchorages available in Auckland."
On Tuesday, the Ministry of Health confirmed two mariners on board the boat - which was off the coast of Taranaki - had tested positive for Covid-19.
The mariners were part of a group of nine who arrived in Auckland on Monday and were then driven to their deep-sea fishing vessel in New Plymouth.
Yesterday Port Taranaki said it would not allow the ship to return to the port - as doing so would put staff and the community at "a higher risk".
Today the Ministry of Health confirmed one of the two infected mariners had the Delta variant of Covid-19 and that genome sequencing showed this infection was not linked to any other cases in New Zealand.
An All-of-Government response had been set up, led by Customs and including health officials, border agencies, port authorities and the shipping company, to address the health needs of crew members and determine the next steps for the vessel.
NZ Customs wouldn't be drawn to comment on the Ports of Auckland's offer to take the ship at its quarantine anchorage - but said once an agreement had been made to find the ship a place to dock, the fishermen would undergo Covid-19 testing and move to an MIQ facility.
Craig Harrison, of the Maritime Union, confirmed that the infected pair were on the Spanish-flagged Viking Bay.
"Our concern now is for the crew," he told Newstalk ZB.
"It wouldn't be much fun [for] the crew on a small fishing vessel - everyone telling them to isolate."
Health officials continued to deem this situation low risk to public health due to the infection prevention control process in place.
There are no associated locations of interest.
Meanwhile, one new Covid case in managed isolation was reported 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.