It's understood that one New Zealander is aboard the Panamanian-flagged vessel, called the Ushuaia after the Argentine port in Tierra del Fuego port from which it sailed on Sunday.
Argentine Navy Admiral Daniel Martin said a nearby passenger ship was on its way to the stricken vessel.
He and the boat's captain, who also spoke to local media, said none of the boat's passengers had been injured and contingency plans were being followed.
"The aim is to get the passengers off as soon as possible and this they're going to do with the Atlantic Dream, which will soon be arriving there," he said, referring to the nearby cruise ship being sent to the rescue.
He said the ship lay some 300 km southwest of Argentina's Marambio military base on the Antarctic Peninsula, in a sheltered strait near Isla Bravante.
Neighboring Chile, which like Argentina has an Antarctic territory, offered to help in the rescue with vessels and helicopters it has nearby.
The Chilean navy said in a statement the crew on the stranded vessel had taken containment measures to prevent the fuel leak from polluting the sea.
Tourism to the Antarctic region has increased five-fold since the early 1990s, as tens of thousands of people cruise during the southern hemisphere summer 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. In that incident people were evacuated and sat in open lifeboats for several hours in freezing temperatures before they were picked up.
But this time it appeared the passengers would be able to await rescue on board the Ushuaia, which is operated by Antarpply Expeditions http://www.antarpply.com/eng/index.php, based in the city of Ushuaia, which lies some 3,200 km south of Buenos Aires.
The company's website says the Ushuaia was originally built for the US National Oceanic and Atmospheric Administration and was refurbished as a cruise ship.