Mr Albanese said the inquiry into the controversy will be headed by former High Court justice Virginia Bell with the expectation that she is due to provide her report by November 25, in time for possible for legislative action before year's end.

Under the inquiry's terms of reference, Justice Bell will examine the facts and circumstances surrounding Mr Morrison's appointments, the implications and the practices and processes surrounding the appointments, including how and where appointments and directions are disclosed. It is expected the Governor-General's role will be examined as Mr Albanese said, "the Governor General, of course, has a particular place under our constitutional system."

The Greens wanted a judicial inquiry with the powers of a royal commission to compel witnesses. Mr Albanese said that was not necessary as Justice Bell's appointment should give confidence to all involved, including Mr Morrison that it was "at arm's length."

"I would find it extraordinary if anyone refused to talk to a former high court judge who is, I think, beyond reproach," he told reporters in Sydney. "I'm certain that this appointment will be seen for what it is and that people will cooperate by giving information, certainly the public service will be required and I don't think it will require compulsion."

"But if it was the case that Virginia Bell felt like she was not getting the cooperation that was required, then I am certain that other measures could be considered."

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Mr Morrison has not explicitly stated his intentions with the new inquiry, but said he would "appropriately assist any genuine process to learn the lessons from the pandemic".

The Opposition leader Peter Dutton has described the inquiry as a "witch hunt" and has criticised Mr Albanese for having an obsession with his predecessor.

The Prime Minister said he regarded this inquiry as "arm's length," with the administration of this inquiry to be done by the Attorney-General's Department, separate to the Department the Prime Minister and Cabinet.

But he particularly points to Justice Bell being "someone far removed from politics who is respected across the board".

"No one can argue that this isn't a former minister or politician being appointed," Mr Albanese said.

"The fact that we have such an eminent person undertaking this inquiry, I think, means that it should be welcomed by all who hold our parliamentary system of democracy dear, or by all the who understand that we can't take it for granted. And we need to protect it."

Mr Albanese has thanked Justice Bell and described her as an "eminent individual with an extensive and diverse legal career".

Politics has been consumed for two weeks by the revelations that Mr Morrison secretly appointed himself to five portfolios - health, finance, home affairs, treasury and resources - during the pandemic.

On Tuesday, the Solicitor-General Stephen Donaghue cleared the former prime minister of breaking the law, finding his appointment was valid.

But he was scathing about Mr Morrison's decision to keep the appointments secret from parliament, his fellow ministers and department secretaries, finding that those actions "fundamentally undermined" the principles of responsible government.

Mr Morrison has defended his actions and stressed that he had only ever used his secret powers once - to kill off the PEP-11 gas exploration permit.