Maritime NZ said the Golden Mind lost steering west of the island last Wednesday, and the ship and its crew had been waiting there since.
In an update this morning, it said the ocean-going tug MMA Vision - which set off from Wellington on Thursday - was now alongside.
However, weather conditions would determine when the two boats could connect, it said.
Weather stations on Rakiura were recording gale-force winds today.
Maritime NZ said the plan was to tow the tanker to Timaru.
"Towage arrangements are made between the commercial parties involved - in this case, Golden Mind's owners and MMA Vision."
Maritime NZ said the crew was safe and no distress calls had been made.
The authority's Rescue Coordination Centre would continue to keep watch but it would not be issuing any further statements about the situation today.
