New Zealand's oil spill response agency, Maritime New Zealand (MNZ), said in a statement that the ship has sustained "some damage from current movement and there is a significant amount of oil leaking from the vessel."
MNZ estimates that 130-350 tonnes of oil have leaked from the vessel, which was carrying 1,700 cubic meters (450,000 gallons) of fuel.
The agency said that the vessel was still intact, but was being moved around by strong seas.
Clumps of oil from the leaking ship have been found on Mount Maunganui beach in the Bay of Plenty.
MNZ is advising people to avoid the beach for health reasons.
Oil began leaking from the Rena, a Liberian-flagged vessel, after it struck the Astrolabe Reef, about 12 nautical miles off the coast of Tauranga, on the North Island, on Wednesday, creating a five-km (three-mile) slick.
Efforts to recover oil from the grounded ship, now heading south down the coast, are being hampered by high winds and strong swells.
Approximately 10 tonnes of fuel oil was transferred to tanks on board a barge before the recovery operation was suspended on Monday.
MNZ said dispersants are being tested on the oil that has leaked. However, there are sea swells of up to four meters, which can make it difficult for the dispersant to work.
Worsening weather prompted the ship's crew to be evacuated as a precautionary measure after a mayday call was made on Tuesday.
Naval architects and salvage experts on board the Rena are monitoring the vessel and have placed sensors to provide advance warning should the ship's structure come under too much stress.
MNZ has established a one-kilometer maritime exclusion zone around the ship due to the health risk posed by the oil spill.
National On-Scene Commander Rob Service said the operation to remove the oil from the ship, led by Svitzer Salvage, would be complex.
"There is no way of doing this quickly," he said.
A rescue center has been set up in the city of Tauranga to clean oil from animals affected by the spill and MNZ has 12 teams working to assist oiled wildlife. Nine birds have been recovered -- seven little blue penguins and two shags, the agency said.
About 300 defense force personnel are on standby for a shore cleanup if needed, the agency said.
Officials fear the stricken ship may spill additional oil as its condition deteriorates.
"I think we'd expect that at some point to have further problems and potentially even break up," Transport Minister Steven Joyce told broadcaster TVNZ.
Greenpeace expressed "extreme concern" about the spill and urged the government to avoid using further toxic dispersants.
"This is an unfortunate illustration of just how difficult it is to deal with oil spills at sea," the organization's Steve Abel said. "Even a slow, and relatively accessible oil spill like this one has clearly stretched New Zealand's response capability to its limits.
"It is also a potential disaster for the blue whales and dolphins presently calving in the area, as well as numerous other marine species."
New Zealand Prime Minister John Key said two inquiries had been launched into the incident to discover how the ship hit the reef.
"This is a very large ship, that in calm waters has hit an extremely well documented reef," he told Television New Zealand.
"Some serious questions need to be answered about why that happened and who is responsible," he said.