The Polar Star had been en route to Antarctica since December 3 to clear a shipping channel in McMurdo Sound to the National Science Foundation's scientific research station.
The Polar Star would go on to do this after the rescue attempt, the AMSA said.
It would leave Sydney today and was expected to take about seven days to reach Commonwealth Bay, depending on the weather and ice conditions.
At 122m, the Polar Star is one of the largest ships in the US Coast Guard fleet. It is able to break ice over 6m thick and can continuously break ice up to 1.8m while travelling at 3 knots.
The scientific team on board the Akademik Shokalskiy, led by scientists from the University of New South Wales, had been recreating Australian explorer Douglas Mawson's 1911 to 1913 voyage to Antarctica when they became stuck.
The six New Zealanders are ornithologist Kerry-Jayne Wilson, University of Auckland doctoral student Colin Tan, historians John and Barbara Tucker, and two chefs.