The former prime minister has not confirmed if he'll participate in the inquiry initiated by Labor that will be run by former High Court justice Virginia Bell and report by November 25. It will examine Mr Morrison's moves to give himself administration control over Treasury, Home Affairs, Finance, Health, and Industry, Science, Energy and Resources portfolios.

The actions were not unlawful according to Solicitor-General advice released by Prime Minister Anthony Albanese last month when he also announced he was setting up the inquiry.

Justice Bell has invited "any interested individual or organisation" to make submissions via the inquiry website by the end of the month. She has encouraged anyone making a submission to refer to the terms of reference, which encompass examining the facts and circumstances surrounding the appointments, their implications, as well as accountability to Parliament and public confidence in government.

Mr Albanese had urged all parties to participate in the inquiry. It would be "extraordinary" if Mr Morrison did not cooperate, he said last month.

The appointment of Justice Bell would give the public confidence in the inquiry as it was being run independently "at arm's length", he said.

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Opposition Leader Peter Dutton said the scrutiny had turned into "a political witch-hunt" after the Prime Minister's announcement. If it turned into a "show trial", the Coalition would withdraw its support for measures suggested by the Solicitor-General.

"Scott did the wrong thing, he admitted that. We've got the facts in relation to the matter," Mr Dutton said. "I think there's a tipping point here for the Prime Minister, and I think frankly, he went too far, and it's turned into a political witch-hunt."