Maritime NZ also confirmed it was aware of the incident and was making initial enquiries.
St John ambulance service had called the fire service at 9.21am with a report a person had been injured, Fire and Emergency NZ northern shift manager Carren Larking said.
Four fire appliances were sent to the site. The fire service was still at the scene but any further request would need to be directed to the ambulance service, Larking said.
Craig Harrison, the national secretary of the Maritime Union of NZ, headed to the port this morning but had not been able to speak to anyone as the site was still blocked off while police investigated.
He said the union was disappointed to hear of another death but could not say much more until it knew what had happened.
The Ports of Auckland's safety record has previously been under intense scrutiny after three deaths since 2017.
The results of an independent health and safety review in April last year identified the need for significant improvements.
The review by Construction Health and Safety New Zealand (CHSNZ) was commissioned by the port's owner, Auckland Council.
It found "systemic problems at POAL in relation to critical Health and Safety risk management and organisational culture that relate to Health and Safety" and "opportunities for significant improvement".
While the port company is wholly owned by Auckland Council, it operates under the umbrella of the Port Companies Act 1988, meaning the council cannot cut across operational management.
But Auckland Mayor Phil Goff said that his ordering of the independent review was an unprecedented step towards addressing safety concerns and said he was demanding accountability from the board of directors, which the council can influence.
The port's chief executive Tony Gibson resigned in May, one month after the review's findings were released.