Talks have resumed between the Ports of Auckland and the Maritime Union following industrial action which has cost the company over two million dollars so far.
More than 300 union members are involved in a dispute over a collective contract.
Union national president Garry Parsloe says the company put a revised offer on the table, which it is considering.
Mr Parsloe says mediation may resume on Friday or next week if a deal is not struck.
He says the union will also review a strike notice for 23 December and Christmas Day, depending on how the talks go.
Ports of Auckland says it is urging the union to withdraw its planned strike action.
Earlier industrial action, including a strike and a lockout closed the container terminal at the port for four days.