Earlier rescue attempts by Chinese and French icebreakers were foiled by the thick ice.
Seventy-four scientists, tourists and crew are on the Akademik Shokalskiy.
The vessel is being used by the Australasian Antarctic Expedition to follow the route explorer Douglas Mawson travelled a century ago.
The Shokalskiy remains well stocked with food and is in no danger, according to the team.
Despite being trapped, the scientists have continued their experiments, measuring temperature and salinity through cracks in the surrounding ice.
The Australian Maritime Safety Authority (AMSA), which is coordinating the rescue, said the Aurora Australis was expected to reach the trapped research ship on Sunday around 12:00 GMT.
The powerful icebreaker can cut ice up to 5.2 ft thick – potentially still enough to plough through the estimated three-metre wall surrounding the Shokalskiy.
If this latest relief operation fails, passengers could be winched to safety by a helicopter on board the Chinese icebreaker, which had to abort its rescue mission on Saturday.
The Snow Dragon came within seven nautical miles of the Russian ship before stalling and being forced to return to the open sea.
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