Calls are growing for an inquiry to be launched into the former Liberal leader's decision to secretly appoint himself as a joint minister in the health, finance, treasury, home affairs, industry, science, energy and resources portfolios between March 2020 and May 2021.
One of his decisions, as joint resources minister, was to block the PEP-11 gas exploration licence off the NSW coast after the then resources minister Keith Pitt approved the project. The final decision is now the subject of a Federal Court appeal.
Mr Albanese has called for reform to ensure similar presidential-style acquisitions of powers can't happen again under Australia's Westminster system.
But Greens leader Adam Bandt wants an inquiry to examine Mr Morrison's conduct and whether he misled the parliament or pressured the governor-general or public service.
"The longer that this goes on, the more questions that are raised about exactly how much damage has been done to Australian democracy," Mr Bandt told ABC News on Monday.
"You're entitled to expect that when the prime minister stands up and says 'here's who the ministers are in my parliament, here's what my responsibilities are', you're entitled to expect that they are telling the truth and it's becoming clear day by day that that's not the case."
The Greens have already asked House of Representatives Speaker Milton Dick to refer Mr Morrison to the powerful privileges committee over the scandal.
Mr Bandt said an inquiry would also need to examine who was asked by Mr Morrison to keep his ministries secret from the public and most of his colleagues.
"One of the things that we don't yet know is whether or not that was done at Scott Morrison's requests, whether or not he made requests of the governor-general to keep some of these appointments secret," he said.
"Similarly, with respect to his own department, what did ... the former prime minister do?"
Coalition MP Bridget Archer said she would also welcome an inquiry into the conduct of the former prime minister.
She said she had an open mind about how an inquiry would be conducted.
"Australians generally agree that we don't want to see this sort of situation occur into the future, so we need to examine how we got here and how we stop it happening going forward," she told ABC Radio on Monday.
Ms Archer said while she was reluctant to ask for other MPs to resign, the former prime minister should "reflect" on his behaviour.
"It also is forcing other colleagues to..kind of having to defend themselves and their own records. And I think that that's unfair," she said.
"(Mr Morrison) should reflect on that and, and certainly consider whether it's the best thing for him to do going forward."
Mr Albanese said Mr Morrison's self-appointments needed to be scrutinised.
"There's separate questions about the functioning of our democracy, about conventions and whether any conventions have been overturned and whether there's a need for any reforms required to ensure that something like this can never happen again," Mr Albanese told Sky News on Sunday.
"Very clearly, there's a need for proper scrutiny of what occurred here, this was an undermining of our parliamentary democracy."
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