The process of contracting out includes distributing a formal "request for proposal" document to five private companies this week to replace the union workforce.
Mr Gibson said the company was still prepared to listen to innovative solutions to work practices from the union.
The port company's final offer last week of increasing hourly rates by 10 per cent in return for replacing eight-hour duties with shifts ranging from five to 12 hours was met with a counter-offer by the union of a 2.5 per cent increase and rolling over the existing contract for six months to investigate ways to boost productivity.
The port company has refused to say what its offer would mean for labour costs at the port, except that its staff hours per container were about 1.21 hours and it would like to reduce that figure to less than 1 hour.
Mr Gibson said the counter-offer fell well short of what was needed to improve work practices and productivity.
Maritime Union president Garry Parsloe said the union was entering mediation to mediate, but Mr Gibson was going in with a take-it-or-leave-it position.
"We hope he can be a little bit more conciliatory because we intend to be. We are saying we have had the strikes, the stoppages, the lockouts and all the yelling and screaming.
"It is now in the interest of the port and the people of Auckland that we sit back and try and listen to each other and come up with a compromise ... that both parties can live with," Mr Parsloe said.
He said the union had tweaked its offer in areas such as a roster system to come closer to what Mr Gibson wanted.
A senior source at the Auckland Council, which owns 100 per cent of the port company, said it was a critical week for the dispute.
Leah Thornton yesterday took her four children, Nina, 4, twins Amy and Max, 5, and Ben, 9, down to support their father at the picket line.
"This is not just about the guys, this is about the union. And this is a family issue, it affects everybody," she said.
Mrs Thornton and her children, among other workers' families, took turns holding the signs which read: "All my Daddy wants is a roster".