The 63-metre (207-foot) "Antarctic Chieftain" became trapped in ice some 900 nautical miles (1,650 kilometres) northeast of McMurdo Sound on Tuesday, damaging its propeller.
The US Coast Guard cutter "Polar Star" was ordered to cross through 720 kilometres of iceberg-strewn waters to the ship's aid.
The New Zealand Rescue Coordination Centre, which oversees the Southern Ocean search and rescue zone, said the US ship, which has a reinforced icebreaker hull, reached the trawler early Saturday after breaking through the ice around it.
"They are making slow but steady progress and are currently approximately 60 nautical miles from clear water."
The Americans had earlier deployed a remote controlled mini-submarine to assess the damage to the "Antarctic Chieftain's" propeller and gauge if it could travel under its own steam.
"The (propeller) blades were assessed as too badly damaged for the vessel to be able to use them for propulsion from the ice field," Reynecke said.
New Zealand rescue coordinators say the trawler's 26 crew members are not at risk and there has been no oil leak in the environmentally sensitive area.
A New Zealand-flagged trawler "Janas" is also travelling to the area to provide assistance if required but is not expected to reach the vessels until late Monday.
The "Antarctic Chieftain", built in 2002, is licenced to trawl for Patagonian toothfish, a slow-growing species that has a type of anti-freeze in its blood to deal with the punishing southern conditions.
