Robertson, speaking on behalf of Prime Minister Jacinda Ardern said: "The inquiry has been set up to look into whether or not all relevant information was available in the appointment of Mr Haumaha. On the face of it, it would appear to me that those allegations appear to fit within that category of the terms of reference."
"Any suggestion of bullying or intimidation in Government departments is unacceptable," Robertson added.
Hipkins, speaking on behalf of Internal Affairs Minister Tracey Martin, who is the commissioning minister of the Government inquiry, reiterated that sentiment to Chris Bishop.
"The terms of reference don't need to be adjusted for that matter to be investigated as part of the inquiry," Hipkins said.
"It is certainly within the scope of the existing inquiry as the terms of reference make very clear."
It was revealed today that three people from the Department of Corrections and Ministry of Justice walked out of a project in 2016 being run from Police National headquarters and refused to return, alleging verbal bullying behaviour towards them.
The project was designed to improve "justice outcomes" for Maori.
Haumaha was a superintendent at the time working in the Maori, Pacific, Ethnic Services division.
One of the analysts involved said the problems arose when others in the group questioned his "police-centric" approach.
"When we raised issues with Haumaha about his police-centric views, we were brushed off and pretty much told 'you're in my building, do as I say,'" one of them told the Herald.
After leaving Police HQ, the analysts, two women from Justice and one from Corrections, continued working on it from the Ministry of Justice.
The Government inquiry into Haumaha's appointment was announced the day after the Herald revealed comments he had made during an investigation in 2004 into historic rape allegations against several of Haumaha's friends.
Haumaha described her allegations as "nonsense" and described one of the accused, Bob Schollum, as a 'legend" and another, Brad Shipton, as "a big softie."
The inquiry will look at whether the appointment panel and cabinet had all the relevant information when making their recommendations and decisions.
It has also emerged that Haumaha has had a close association with New Zealand First, which was consulted by Nash over the appointment, and was once named as a candidate for the party before withdrawing.
The cabinet is still considering candidates to conduct the inquiry after the original reviewer, Pauline Kingi, resigned last week amid revelations by the Herald that she had endorsed Haumaha 23 times on the professional networking website LinkedIn.
