Mr Service said the ongoing fuel oil leak was intermittent and appeared to be coming from damaged pipework on the vessel.
"We are not aware of any actual breaches in the fuel tanks. However, because of the extensive damage to the vessel, it is difficult to determine accurately what the scale and scope of the damage is."
It was not yet known how much oil had spilled from the ship and the crew were working to prevent further leakage.
The Rena was carrying about 1700 tonnes of heavy fuel oil and about 70 tonnes of marine diesel when it ran aground.
An oiled-wildlife response team sprang into action today after reports of oil-covered birds in the water around the ship.
The reef - about 4 nautical miles north of Motiti Island and 12 nautical miles off the coast - is home to wildlife including little blue penguins, seals and petrels.
It was not yet known what species the four dead birds were.
Wildlife response team members would stay on Motiti Island overnight, with beach searches to begin tomorrow.
Maritime New Zealand spokeswoman Sophie Hazelhurst said the ship had suffered significant damage and was in "a difficult position".
"It's quite vulnerable in that it is wedged very firmly on the reef, from the middle of the ship through to the front. The back half of the ship is sort of hanging out over some quite deep water."
Weather conditions were benign with a slight swell, but if conditions changed that could put a lot more pressure on the already significantly damaged ship.
"So yes, we do have concerns for the vessel," Ms Hazelhurst said.
"We're conscious that it is a very damaged ship and we just have to monitor and be ready to respond if things do start to go wrong."
Concerns have also been raised over a hazardous substance aboard the ship that could give off highly flammable hydrogen gas if it came into contact with water.
Maritime New Zealand director Catherine Taylor today issued the ship's owners with two notices under the Maritime Transport Act.
She declared the ship hazardous and instructed its owners to appoint a reputable salvage team.
The owners have selected international salvage company Svitzer to manage the operation, but Maritime New Zealand still has the power to take control of the operation if it is deemed necessary.
They must regularly update Maritime New Zealand on the salvage operation, both before a plan is put into action and once it is underway, and comply with all directions and inspection requirements.
The salvage team will assess damage to the ship before putting together a salvage plan, but it is not yet known what that will involve or how long it might take.