Oil leaking from the Liberian-flagged Rena has created a 5km (3 mile) slick.
An all-out effort is under way to remove nearly 2,000 tonnes of oil from the vessel, which is stranded 12 nautical miles off the coast.
Heavy swells and gale-force winds are forecast for the area from Monday.
Officials says 20-30 tonnes of oil have already spilled into the Bay of Plenty, one of the country's top tourist destinations, since the MV Rena ran aground on Wednesday.
If the ship breaks up, it could release 1,700 tonnes of heavy fuel into an area that is home to whales, dolphins, seals, penguins and a variety of other birds.
'Working tirelessly' John Key, who flew over the scene in a helicopter on Sunday, said two inquiries to determine why the ship had collided with the Astrolabe Reef were already under way.
"People know about the reef, and for it to plough into it for no particular reason - at night, in calm waters - tells you something terrible has gone wrong and we need to understand why," he told Radio New Zealand.
Four naval vessels and two barges are in the area to try to contain the spill.
The barges have already begun scooping up the oil that has spilled into the sea, the BBC's Duncan Kennedy reports from Australia.
A third ship is alongside the MV Rena and attempts are under way to hook them up so the oil can be transferred off the stricken ship.
But the process of linking up the two ships is tricky in the calmest of conditions, and will be all but impossible in gale-force winds and heavy swells, our correspondent adds.
And, because one end of the ship is stuck on the reef while the other floats free, the oncoming storm brings another fear for rescue teams.
"Once you increase the swell, the upsurge, the lift on the aft end of the ship will increase, and the chances of her separating start to rise," explained Marine Risk Assessor John Riding.
"Minimising any impact to New Zealand's coastline is the absolute priority," the firm said in a statement. "The current primary focus of the salvage operations is the safe transfer of the vessel's fuel oil from her tanks."
Our correspondent says that the next 48 hours will determine whether this incident can be contained or become a serious environmental disaster.
Exclusion zone
The department of conservation has established two wildlife rescue centres and dispatched teams to scour the beaches and islands of the Bay of Plenty looking for oil-covered animals and birds.
The maritime authorities has said a total of eight oiled birds, including little blue penguins, had been recovered and taken to a wildlife facility in Te Maunga.
"From tip to toe, they are covered in black sticky gunk, matting up all their feathers right down to the skin," said Brett Gartrell, of New Zealand's Wildlife Health Centre. "They have ingested it and started to get anaemic which is part of the toxic effect of the oil."
Officials say they are preparing for the possibility the slick will hit the coast, after dispersants sprayed from aircraft proved ineffective.
"It has the potential to be very, very serious indeed, simply because of the age of the ship, the damage she's sustained," Andrew Berry of Maritime New Zealand (MNZ) said.
MNZ has established a one-kilometre maritime exclusion zone around the ship and warned that the fuel oil is toxic.
The animal welfare group Forest and Bird said the timing of the accident, in the middle of the breeding season for birds, was "disastrous".