Most voters support recognising indigenous Australians in the constitution and creating a voice to parliament, according to a new poll.
A treaty with indigenous Australians is also strongly backed, the Essential survey confirms.
Indigenous Australians Minister Ken Wyatt has committed to holding a referendum on constitutional recognition within the next three years.
But he and the prime minister will not support a constitutionally enshrined indigenous voice to parliament, as proposed in the Uluru Statement from the Heart.
Conservative Liberals and Nationals have raised concerns the indigenous advisory body could become a "third chamber" of federal parliament.
"We're not in favour of a third chamber or a separate voice," senior minister Peter Dutton told the Nine Network on Friday.
"We've got a very strong democracy, we want to see more indigenous people in the parliament, and it's great that Ken Wyatt is the first (Aboriginal) indigenous affairs minister.
"He's got a process that's under way, let him conduct the consultation and then we'll make an announcement about the next step."
As he tries to build consensus for constitutional change, Mr Wyatt is seeking to calm his colleagues.
"It never was a third chamber," he told The Sydney Morning Herald.
"It is about people, communities wanting to be heard."
Opposition Leader Anthony Albanese described suggestions of a third chamber as "spin".
"This is spin in order to argue against the Uluru Statement From the Heart," he told the Nine Network.
What it simply said was that there is a need for an indigenous voice to parliament, without any power to determine legislation.
"It is not a third chamber. It simply says that where issues affect first nations people, Aboriginal and Torres Strait Islanders, they should be consulted. It is as simple as that."
The proposal for an indigenous voice to parliament - a key recommendation of the 2017 Uluru Statement - has been a vexed issue for the coalition government for years.
Prime Minister Scott Morrison will reportedly veto any move to enshrine an indigenous voice in the constitution.
As such, Mr Wyatt is determined to separate the voice from constitutional recognition of indigenous Australians.
Above all, he is adamant the proposal put to the Australian people must have a strong chance of success.
"If we fail in any recognition in the constitution then we have a problem, because it won't be resurrected for a period of time," Mr Wyatt told ABC Radio National.
"When a constitutional (referendum) fails, then it leaves an impact for our people, it will be a significant impact on the psyche."
Nationals leader Michael McCormack has further muddied the debate, insisting a referendum on constitutional recognition for indigenous Australians should also include a question on recognising local government.
But the deputy prime minister also appears to be kicking the can down the road.
"We won't be having a referendum until the next election," he told reporters in Sydney.