White met with the four councillors who had been surveilled on Wednesday last week and then with all councillors on Friday to discuss the matter.
"What we came out of that [with] is that we would appoint an independent review to look at the concerns that have been raised by the councillors around the scrutiny of their social media and any application for the Code of Conduct."
Councillors declined to comment to RNZ about the inquiry, saying they had been advised not to for legal reasons.
White said the council would be in a position to say more once she had briefed chief executive Wayne Jack about the inquiry and the terms of reference were agreed to upon his return from holiday this week.
"I have refrained from making public comment because of the fact the chief executive is currently returning from overseas and as we are his employer I would like to discuss the matter with him on a face-to-face basis," White said.
While councillors were reluctant to comment on the inquiry, Massey University local governance expert Andy Asquith welcomed it - albeit cautiously.
"My immediate concerns would be who is going to conduct the reviews, what will be the terms of reference and what questions will they be asking?"
A person had in fact been appointed to carry out the task but they could not yet be named, White said.
"The name of the person has been provided to all parties concerned and there has been no concerns raised around the person selected," she said.
A Massey University study in 2014 found Napier was only one of a few councils that prevented elected members from voicing negative opinions publicly.
Study author Catherine Strong said the whole problem stems from an "erroneous interpretation of the code".
"There is a very murky sentence in it that could be interpreted to say that councillors shouldn't criticise council at all, which of course is not the intention.
"Of course councillors should criticise ... they're representing ratepayers, not spin doctors for council," Dr Strong said.
A second independent review looking at the culture of Napier City Council as an organisation had also been called for by some councillors, and while it was considered it was decided it would left for the incoming council to tackle, White said.