"They should stand her down and start again," Mitchell said.
Nash told Newstalk ZB that Haumaha was never going to be the next police commissioner but was "doing a fantastic job in terms of engaging and building relationships between the Government and Māori".
"This is not a review or a look into Wally or his suitability. This is just a look at the process that led to the decision," he said.
"Let's just wait and see how this plays out."
Kingi was questioned by the chief executive of the Department of Internal Affairs yesterday after the Herald contacted the office of Internal Affairs Minister Tracey Martin who is overseeing the inquiry.
The Herald then revealed that Kingi, who is involved in selecting senior police officers, endorsed Haumaha on LinkedIn 23 times.
"She did confirm that she had, like many New Zealanders, set up a LinkedIn account when it was first launched," Martin told Parliament yesterday in reply to questions from National MP Chris Bishop.
"At that time it was common practice for Māori professionals to support each other through this new medium through endorsements."
Martin revealed Kingi had declared she knew of Haumaha in a professional capacity and also attended the same tangi as Haumaha in 2015 or 2016.
"Dr Kingi has signed a declaration saying she has no conflict of interest in the appointment."
Martin expressed full confidence in the process which led to Kingi's appointment, as well as her suitability for the role, citing her "substantial CV" which contains being made a Companion of the New Zealand Order of Merit.
Bishop had called for Kingi to stand down and for Martin to appoint a new chair.
Kingi, a well-respected public servant, was last week announced as chairwoman of the $150,000 inquiry to "examine, identify and report on the adequacy of the process".
She was appointed more than three weeks after the Herald revealed comments made by Haumaha during Operation Austin, an investigation into historic police rape allegations made by Louise Nicholas.
Kingi has been asked to begin her inquiry on August 6, and report within six weeks.
Key terms of reference will include whether all relevant information was properly provided to, or gathered by, the State Services Commission, which has a key role in the appointment of senior public sector roles. It will also look into whether the commission considered all relevant information gathered, or received, to reach its recommendation and whether it provided ministers with all relevant information it had or knew about.