Ports of Auckland bosses are going through a proposal from the Maritime Union on how to end their contract dispute.
More than 300 ports staff went on strike last week, upset at a contract offer they say slashes many established conditions which have been built up over many years.
Ports of Auckland made their fourth offer earlier this month, which included four collective employment agreements to replace a single agreement, an offer union negotiator Denis Carlisle said was "an insult to our intelligence".
A Ports of Auckland spokeswoman said the company had received a counter-offer from the union on Tuesday following mediation.
She said it would take some time to fully examine the counter-offer and an answer would probably be given to the union some time next week.
Ports of Auckland managing director Jens Madsen last week said the company had bent over backwards to come up with an inventive solution from which it could reach an agreement.
The Ports of Auckland, the country's largest container port, is fully owned by Auckland Regional Holdings, the commercial arm of Auckland Regional Council.