The vessel, called the Ushuaia, is adrift but is in no danger of sinking, the Argentine navy has said.
However, reports said the ship was adrift in "difficult conditions".
Admiral Daniel Martin sent out an alarm after the ship suffered a crack.
He said the vessel had run aground in Wilhelmina Bay, part of a peninsula that reaches towards the southern tip of South America.
A nearby Chilean passenger ship is on its way to rescue the 89 passengers and 33 crew.
"We've received information from the captain of the Ushuaia that the boat is grounded ... with a minimal amount of water coming in and some fuel loss," he said.
He said the ship lay some 186 miles (300 km) southwest of Argentina's Marambio military base on the Antarctic Peninsula.
Head of the Argentine National Antarctica Directorate, Mariano Memolli, said those on board were "not in danger". The passengers include Australians and Americans and it is not known if any British tourists are onboard.
Reports said the group would remain on the ship over night.
The Panamanian-flagged ship is operated by Antarpply Expeditions, based in Ushuaia.
Cruise travel has grown in Antarctica in recent years, with tourists paying thousands of dollars to see towering icebergs, seals, whales and penguins.
A year ago, more than 150 crew and passengers, many of them elderly, escaped unhurt in a dramatic rescue after their cruise ship hit ice off Antarctica and sank.