Maritime New Zealand, the agency overseeing the response to the wreck, said up to 350 tons of oil have leaked into the ocean and "significantly more oil" would wash ashore in Papamoa, The Dominion Post reported Wednesday.
Salvage teams were trying to keep the back of the vessel on the Astrolobe reef off Tauranga, where it has been stuck since last Wednesday, in hopes of preventing the 47,000-ton ship from breaking.
The Post said an aerial view from a flight showed a rip on the port side of the ship was worsening.
Meanwhile, the ship's captain has been charged with "operating a vessel in a manner causing unnecessary danger or risk," which carries up to a year in prison and a $10,000 fine. More charges may be filed, the Post said.
The captain, who was not identified, was released on bail.
Clumps of oil have been found along the beaches of Mount Maunganui and Papamoa and a thick oil slick has coated the shore of Motiti Island.
Cargo containers also have washed up after as many as 70 of them slipped from the ship.
"It's devastating to watch," said Motiti Island resident Lynda Wikeepa. "Everyone is out on the point to see the oil just washing up on the rocks. They're almost in tears."
The Post said the Rena is covered for up to $5 billion in insurance claims, a fifth of that for pollution liability. The ship's owners, Costamare Inc., are part of the world's largest group of ship insurers, the newspaper said.
The New Zealand government said it would try to recover costs for environmental damage from those responsible.
Environment Minister Nick Smith called the leak the country's "worst maritime environmental disaster."
Drew Lohrer, a National Institute of Water and Atmospheric Research marine ecologist, said the oil was hitting the coast much faster than bacteria could break it down.
"In areas where the oil arrives in thick slicks or clumps, it may take years or decades for it to disappear naturally," Lohrer said. "This is why it is imperative to clean up as much of the spill as possible."