Stricken cargo ship MV Shiling is under tow and due to arrive in Wellington Harbour on late on Tuesday morning.
The Shiling put out a mayday call, when it lost power near Farewell Spit on 12 May. It has since been anchored in Tasman Bay.
Ocean-going tug Skandi Emerald began the towing operation on Monday morning.
Maritime New Zealand incident controller David Billington said the weather conditions were looking stable for the journey.
The tug's crew were experienced in managing tows in a range of conditions, including those typical of the Cook Strait, Billington said.
The vessels are currently tracking at between 5 knots and 6 knots.
Maritime NZ's incident response team will monitor the vessels as they make their journey to Wellington.
All of the Shiling's 24 crew remain onboard the vessel.
Billington said ensuring the operation was undertaken as safely as possible was the top priority.
The vessels are due to arrive in Wellington about 11am on Tuesday.
A 500-metre exclusion zone has been put in place around the towage operation for the duration of the journey to Wellington.
Wellington Harbourmaster Grant Nalder said the cargo ship had a problem with its main engine and its crew had done assessments, with technical support and spare parts on the way to Wellington.
It was not known how long it will remain at CentrePort.
Nalder said the Shiling was berthed in Wellington for three weeks earlier in the year while repair work was done.