The announcement followed a third failed attempt to free the cruise liner - carrying 90 Australians - after a fisheries research vessel attempted to pull the ship free at high tide on Wednesday.
"These passengers are currently in isolation," the Australia-based Aurora Expeditions said in a statement on Thursday.
"They are looked after by our onboard doctor, medical team and crew, and they are doing well."
The others on the MV Ocean Explorer were "safe and healthy", it said.
The Sydney Morning Herald quoted a retiree from Australia, Steven Fraser, who is on the ship, saying "everyone's in good spirits".
"It's a little bit frustrating, but we are in a beautiful part of the world," he told the newspaper, adding he had come down with COVID-19 on the ship.
The cruise ship ran aground above the Arctic Circle on Monday in Alpefjord, in the Northeast Greenland National Park - the world's northernmost national park.
The park is almost the size of France and Spain combined, and approximately 80 per cent is permanently covered by an ice sheet.
Alpefjord sits about 240km away from the closest settlement, Ittoqqortoormiit, which itself is almost 1.4km from the country's capital, Nuuk.
"Unfortunately, the attempt (to free the ship) was not successful," said the Danish Joint Arctic Command, which is co-ordinating the operation.
Earlier this week, the cruise ship made two failed attempts to float free on its own during high tide.
In a statement, the Arctic Command said its "first priority" was to have its larger inspection vessel Knud Rasmussen reach the site, saying the ship was expected on Friday evening after it had to "slow down a bit" on its way because of the weather.
The Bahamas-flagged cruise ship has passengers from Australia, New Zealand, South Korea, the UK and the United States.
It has an inverted bow, shaped like the one on a submarine, 77 cabins, 151 passenger beds and 99 beds for crew, and several restaurants.
The Arctic Command earlier said there were other ships in the vicinity of the stranded cruise liner.
Members of the Sirius Dog Sled Patrol, a Danish naval unit that conducts long-range reconnaissance and enforces Danish sovereignty in the Arctic wilderness, were also nearby.
The latter visited the ship on Tuesday and reported everyone on board was fine and no damage to the vessel was reported.
The weather for the next few days shows sun, blue skies and temperatures of about 5C, according to the Danish Meteorological Institute.
Greenland newspaper Sermitsiaq said police in Greenland were investigating why the ship had run aground and whether any laws had been violated.
So far, no one has been charged or arrested.
The daily cited a police statement saying an officer had been on board the cruise ship to carry out "initial investigative steps, which, among other things, involve questioning the crew and other relevant persons on board".
The primary mission of the Joint Arctic Command is to ensure Danish sovereignty by monitoring the area around the Faeroe Islands and Greenland, including the Arctic Ocean in the north.
Greenland is a semi-independent territory that is part of the Danish realm, as are the Faeroe Islands.
19°C