The ice-bound ship, the Akademik Shokalskiy, left New Zealand on Nov. 28 on a privately funded expedition to commemorate the 100th anniversary of an Antarctic journey led by famed Australian explorer Douglas Mawson.
It has been stuck in the ice since Dec. 24. Its passengers include scientists and tourists, many of them Australian, and a Russian crew.
The Australian icebreaker the Aurora Australis is expected to reach the stricken ship at about midnight on Sunday.
A Chinese icebreaker could not break through the thick ice earlier but the weather on Sunday boded well for a rescue.
"The ice conditions seem to have improved and there appears to be some softening and some cracks appearing," Lisa Martin of the Australian Maritime Safety Authority, which is coordinating the rescue, told Reuters.
Just how the rescue would be done would be worked out when the Aurora reached the area, she said.
Those on board were in good condition and have never been in any imminent danger.
"We're primarily looking to the Aurora to get us out," Chris Turney, an Australian professor onboard the beleaguered ship who is leading the expedition, wrote in an email to Reuters on Sunday.
"Hopefully there are some breaks developing in the surface from the weaker winds and sun during today."
The ship is stuck about 100 nautical miles (185 km) east of the French Antarctic station Dumont D'Urville and about 1,500 nautical miles (2,800 km) south of Hobart, Tasmania.
The Aurora is the third icebreaker seconded by the Australian maritime rescue agency to try to reach the hemmed in ship.
The Chinese icebreaker, the Snow Dragon, is on standby at the edge of the ice and within sight of the trapped ship. It has a helicopter on board which could be used in the rescue.
A French icebreaker had also tried to help.