Only one ship, the Australian Antarctic Division icebreaker Aurora Australis, is still making its way to the MV Akademik Shokalskiy.
It was expected to arrive later this evening (Sunday).
"We don't expect it to arrive until quite late this evening Australian eastern standard time,'' said Australian Maritime Safety Authority spokeswoman Lisa Martin.
"So in terms of getting those people off, nothing is really going to happen in the next 12-24 hours.
"It is making good progress and once it arrives ... it will assess whether it is capable of making its way to the Akademik or whether we need to look at other options.''
Snow Dragon, which has a helicopter on board, has remained in the area to help with a rescue if required.
It could transfer some people on to its own vessel, and others on to the Aurora Australis, Ms Martin said.
Aurora Australis can accommodate up to 80 additional people and has large supplies of food and fuel.
Where the people would be taken once it was lifted off the ice was yet to be determined.
That decision would be made by the ship's master, in consultation with the Australian Maritime Safety Authority, Ms Martin said.
"We won't be able to say until we get them on board and assess their needs.''
The MV Akademik Shokalskiy left Bluff on December 8 on an expedition led by scientists from the University of New South Wales.
The stranded ship was retracing Sir Douglas Mawson's 1911 Antarctic expedition when it became trapped.
Among the stranded passengers are six New Zealanders - ornithologist Kerry-Jayne Wilson, University of Auckland doctoral student Colin Tan, historians John and Barbara Tucker, and two chefs.
RCC (Rescue Coordination Centre) Australia is in regular contact with the MV Akademik Shokalskiy and the 74 people on board are reported to be safe.
They have reportedly been in high spirits while awaiting rescue.
Janet Rice, an Australian Green party politician who has been on board since the ship left New Zealand, told a Guardian newspaper journalist who is also on board: ``I understand why people might be concerned, but the feeling today on board the ship is like a summer holiday when the weather is bad, when you're stuck inside reading books and playing Scrabble. We've been assured that we're in no danger and it's just a matter of waiting.''
