It is understood hospital managers have considered cutting bed numbers to reduce the burden on staff. Te Whetu Tawera currently has capacity for 58 acutely unwell people.
PSA spokesman Brendon Lane said hospital management had offered a deal to resolve concerns around staff safety, workload and assaults by patients.
He said members would vote on a "short-term measure'' which would look at the "flow of patients across the region and how they might be managed across the region".
An Auckland hospital spokesman said it was "working positively with the union" and would not detail the discussions.
A spokeswoman for Counties Manukau District Health Board said there was no final internal or cross-DHB decision on sharing the load.
A spokesman for associate health minister Jonathan Coleman said he had assurances a recent review had studied issues of staff safety. He said the service was "functioning at a high standard".