Finance Minister Grant Robertson says Treasury Secretary Gabriel Makhlouf has not offered his resignation.
This comes after Prime Minister Jacinda Ardern yesterday repeatedly dodged questions as to whether Makhlouf had offered to fall on his sword.
Although Makhlouf finishes his tenure as Treasury Secretary next month, National Leader Simon Bridges had called for his resignation after it was revealed that Budget-sensitive information could be accessed in advance by simply searching the Treasury's website.
Makhlouf will take over as the governor of Ireland's central bank in September.
A spokeswoman for the Irish Minister for Finance Paschal Donohoe told the Irish Times the controversy would not affect Makhlouf's appointment as governor of the Central Bank of Ireland, adding that it was a "matter for the New Zealand Treasury" to deal with.
Robertson said Makhlouf had not offered his resignation over the saga, but he told media the Treasury Secretary had apologised for the blunder on Tuesday night.
"He said he was very sorry this had happened."
Robertson told RNZ this morning that Treasury informed him on Wednesday night that Police had advised them nothing unlawful had taken place.
At 6pm on Tuesday, Treasury contacted the police to report its systems had been "deliberately and systematically hacked".
He informed Robertson that Treasury had gone to the Police just over an hour later.
At 5am on Thursday, Makhlouf said in a statement the Police had advised that someone had "exploited a feature in the website search tool but that this does not appear to be unlawful".
The word hacked was not mentioned in the statement.
This morning, Robertson reiterated his disappointment that the Treasury "didn't do more to find out how this happened before they went to the police".
"He [Makhlouf] came to me and he told me there had been 2000, he used the word hack, attempts to hack into the Treasury system.
"I asked him if he knew how that had happened, he said he did not know at that time how that had happened."
In his own statement at 8pm on Tuesday night, Robertson said he had contacted National to request that they do not release any further material, "given that the Treasury said they have sufficient evidence that indicates the material is a result of a systematic hack and is now subject to a Police investigation".
Bridges said Robertson had clearly implied National had hacked Treasury, or had received hacked information.
Asked if he was being misleading by using the word hack, Robertson said: "All I could rely on was the advice he [Makhlouf] gave me on Tuesday night".
Asked what he thought about National releasing the information, he said he would leave up to New Zealand the "morality and ethics of that".
But he did offer this analogy.
"If you're walking past your neighbour's house and the door is open, it doesn't entitle you to go in and take the contents of the fridge.
"But that's up to individual politicians to decide what they do in those circumstances and New Zealanders will make their own minds up about that."
A spokesman for the State Services Commission (SSC) confirmed an inquiry would look into Treasury and issues around its website security.
SSC boss Peter Hughes was considering the allegations made by Bridges around whether Makhlouf smeared National, or misled the Minister of Finance.
But, at this stage, the spokesman said that was not part of the official inquiry.
"Clearly on this issue of the security of their system, that's something that we now need to look into," Robertson said.
"If you're asking me to express confidence in that aspect of it, well no – I'll wait for the inquiry. But overall the Treasury does a good job for New Zealanders."
A spokesperson for the Treasury said Makhlouf was not doing any media interviews today.
He would not answer questions yesterday.