Labor leader Anthony Albanese said he does not want there to be any delay beyond what is needed to ensure the Constitution is amended to recognise Indigenous Australians through the mechanism of a Voice to Parliament.
The federal government has favoured a process that would establish an Indigenous advisory body that does not require the Constitution to be changed, which would require a referendum.
Treasurer Josh Frydenberg said discussions on whether a referendum should be held on changing the Constitution still 'has a long way to go'.
Mr Albanese rejected suggestions the Voice to Parliament would be a third chamber for parliament, saying the body would allow First Nations people to have a say in policies that affect covering health, education and housing.
He reaffirmed his within the first term of an Albanese Labor government.
"This is a change that has been a long time coming. We've been talking about it since at least the end of last century," Mr Albanese told the ABC's Insiders program.
"I will consult with First Nations people about the timetable. I will reach out across the parliament ... to try to secure support as much as possible."
Labor Leader Anthony Albanese during a press conference in Darwin. Source: AAP / LUKAS COCH/AAPIMAGE
"This is a nation-changing moment. Just as the apology to the Stolen Generations made our country stronger, this is a generous offer for First Nations people," he said.
Deputy Liberal leader and Treasurer Josh Frydenberg said constitutional recognition is important, but it is not the focus of the Coalition's approach.
"That's been a consistent position of not just this prime minister but a previous prime minister," he told the Insiders program.
He said what the government has been focusing on are "regional voices", which he said is the aim of the government's non-constitutional voice to parliament.
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Legislation has yet to be enacted to deliver on the plan.
"That's what (Indigenous Minister) Ken Wyatt has done a lot of work (on) and received support from the local Indigenous community to do so," he said.
"We have put significant funding in the budgets in order to focus on those regional voices."
Asked directly if he supported a Voice to Parliament in the Constitution, Mr Frydenberg said: "I think this debate has some way to go."
"I do see some challenges with it and ultimately Australia would be best served by having a bipartisan approach on this, but we do support constitutional recognition of our First Australians."
Indigenous Australians Minister Ken Wyatt with members of the Kakadu Aboriginal Land Trust at a land handback ceremony. Source: AAP / AARON BUNCH/AAPIMAGE
The First Nations delegates proposed two possible dates for a referendum to be held on the issue: 27 May 2023 or 27 January 2024.
"The politicians were not ready for Uluru in 2017. But now the Australian people are.
"We call on all sides of politics to support our call."
More than 250 Aboriginal and Torres Strait Islander leaders backed the Uluru Statement of the Heart after months of consultation.
The statement calls for 'First Nations Voice' in the Constitution and the creation of a Makarrata Commission for treaty-making and truth-telling.
