It is unclear what impact this will have on the sailing, which was scheduled to be at sea today.
White Island has been a popular day-trip destination for both tourists and cruise passengers because of its dramatic volcanic activity.
However, the future of tourism on Whakaari is in doubt.
Last Tuesday GeoNet warned of a heightened level of volcanic activity but that it did "not pose a direct hazard to visitors".
As well as Royal Caribbean, ships from Azamara and Celebrity cruises have previously offered excursion to the island.
Princess Cruises - which sails three ships via the Port of Tauranga - offers scenic helicopter overflights to White Island, though passengers do not land on the island.
Operator White Island Tours which offers day trips to "explore New Zealand's most active volcano" has taken its website down following the emergency.
Passenger Venessa Lugo opted not to go to White Island. Photo / George Novak, File
Passenger Venessa Lugo opted not to go to White Island. Photo / George Novak, File
"It was one of those things you just stand there in shock. What are the chances this happens on your holiday," passenger Venessa Lugo, 24, told the Herald.
Lugo and her partner opted to go on an excursion to Hobbiton yesterday, instead of the White Island tour.
Passengers were informed of the disaster by an announcement from the ship's captain this morning.
"He said there was a tragedy that occurred and that we had missing passengers and crew from the volcano," said Debbie Shapirl, who is cruising with family.
Donna Field a reporter with ABC news who was holidaying on the Ovation tweeted a picture of the ship's upper deck, saying : "Bit surreal onboard this morning."
Locals have been leaving flowers and tributes at the Port of Tauranga, where the Ovation is still berthed.
The Ovation is due to conclude its sailing in Sydney on Monday 16 December.
At full capacity she has just under 5000 passengers and 1500 staff, making her the largest Cruise ship ever to visit New Zealand.
Family members concerned about relatives on board Ovation of the Seas can contact the Royal Caribbean cruise line on these numbers:

Australia – 1300-026-240
New Zealand – 0800-002141
USA – 800 829 4050
Germany 0800-180-0885
UK 0-800-014-8339
Mexico 01-800-681-5336
China 400-120-3534
Japan 0800-170-6282
Singapore 65-31582855