In a response via a written statement, the company said "Mr Gibson has the board's full support".
In the same statement, the port company denied it leaked to the media Jones' warning to Gibson at the meeting "do not put your head in a political noose", and accuses Jones of "bullying and threatening behaviour" which is "unprofessional and unbecoming of a minister".
"We do not tolerate bullying in the workplace nor should we have to tolerate it from Shane Jones."
Meanwhile former Labour prime minister Helen Clark has weighed into the controversial proposal by a Government-commissioned working group that the Ports of Auckland should be closed to all but cruise ships and deepwater port Northport be developed. The working group's final report, still with Cabinet, is believed to conclude the Auckland ports are not viable either environmentally or economically.
Clark overnight tweeted: "It's a big vision & would be a win/win for Northland & Auckland. My humble opinion as an Aucklander for more than 5.5 decades....
Total game changer. Auckland can be a world class city & a compelling vision for the waterfront is part of that - Ports of Auckland's car park isn't.
Much to be said for big vision for upper North Is port infrastructure: regional development/employment benefits for Northland, + could realise benefits for #worldclasscity development for Auckland; better than car park vision of #POAL for Akld waterfront".
During the "robust" meeting in Robertson's office, attended by transport minister Phil Twyford, Jones, Robertson, Gibson and Government officials, Jones said he advised Gibson "this is as much a political matter as an economic matter and I advise you, Tony, do not put your head in a political noose".
"(I said) That's not your role that's Phil Goff's role."
Goff, mayor of Auckland, is on holiday in Australia and not available for comment.
Jones said Gibson leaked the exchange to the media.
"I don't mind businessmen trying to run a business but once you start leaking and briefing media then you've stopped being a businessman and stepped into politics. You stand to reap what you sow."
The Ports of Auckland statement said the leak wasn't from the company and rejected Jones' claim that the company and Gibson had started a political campaign to undermine NZ First. The party made a study of the future location of upper North Island ports a requirement of its signature to the Coalition agreement when a new Government was formed in 2017. Jones and NZ First leader Winston Peters both come from Northland.
"We haven't, we are simply putting the facts in the public domain. We believe this debate needs to be about facts not politics," said the statement.
The ports company this week released two reports by economic consultancies questioning the robustness of an economic analysis by Ernst & Young supporting the working party's interim recommendation that Auckland port operations be relocated north.
The company said the two reports found "major problems" with the analysis.
Asked if questions around the legality of NZ First's use of funds from its mysterious NZ First Foundation could damage the party's ports reform effort, Jones said the two were separate issues.
"One pertains to the Electoral Act, the other pertains to cowboy behaviour from the Ports of Auckland."
Jones wouldn't comment further on the NZ First Foundation.
