Republicans want the appointment of the next Governor-General to be a trial run for choosing an Australian head of state.
This could involve a public vote, nominations moved by parliament or a parliamentary vote, Australian Republic Movement chairman Peter FitzSimons wrote in a letter to Prime Minister Malcolm Turnbull.
Mr Turnbull was a founding member of the ARM.
National director Michael Cooney says the movement wants a national vote on questions of an Australian head of state in 2020 and a referendum to change the constitution in 2022.
"We believe Australians from culturally and linguistically diverse backgrounds, women and Indigenous Australians should have an equal opportunity to serve," he said.
Labor leader Bill Shorten suggested on Sunday any move toward a republic should happen alongside a treaty with indigenous Australians.
"I think it's remarkable that over two centuries after first European settlement we are still borrowing a very worthy person but a monarch from another country," he told Sky News.
But the Australian Monarchist League warned Mr Shorten was making a "huge blunder".