Putting an Indigenous advisory body in the constitution would be the "ultimate" way of giving Australia's first peoples a "fair go", according to Labor leader Anthony Albanese.
A major conference at Uluru two years ago proposed an Aboriginal and Torres Strait Islander voice to parliament.
The group would advise MPs and senators on laws and issues affecting First Nations people.
Mr Albanese on Saturday delivered his first major Indigenous affairs speech since becoming Opposition Leader at the Garma Aboriginal festival in Arnhem Land.
"Until it exists, we are all diminished," Mr Albanese told the event.
"It would be the ultimate fulfilment of that most Australian of instincts: the fair go.
"More than anything, it would be right."
Minister for Indigenous Australians, Ken Wyatt, reignited debate about the Uluru Statement last month when he suggested a referendum could be held within three years.
But, the idea of enshrining the voice in the constitution has proved controversial within parts of the Coalition.
Prime Minister Scott Morrison has rejected the idea but suggested the body could be legislated instead.
Critics have suggested the body could threaten the sovereignty of the parliament, while others are wary of any changes to the nation's founding document.
But Mr Albanese urged sceptical Government MPs to "give this a chance", saying the men who wrote the constitution expected the document to evolve.
"They did not set out to create a museum piece to be put behind a piece of glass and an ever-thickening layer of dust," he said.
"They created a living document. They designed the constitution to be enduring, but not unchanging."
Mr Morrison last month insisted the voice was "not being considered within the constitutional context" but would instead be legislated through the parliament.
Mr Wyatt said he would "find other solutions", saying Mr Morrison was "committed to seeing a difference made in respect of this issue."
"If it's not going to be entrenched, then we don't entrench it," Mr Wyatt said.
Mr Wyatt, who is a Yamatji-Wongi-Noongar man, said he would gauge the support of colleagues for change.
"I need to know who are our strong supporters, who are the people who want clarity and the people who will indicate that they don't support it," he said.
Governor-General David Hurley, who attended the opening of the festival on Friday, said work towards a referendum needed to progress quickly.
"We're going to have … a referendum in this term," he said.
"Three years in the cycle of the country is a very short time.
"So we need to move fairly quickly on what we think [are] going to be the outcomes here."
General Hurley has previously supported the teaching of Indigenous languages in schools and trained with Aboriginal boxers in inner-Sydney.
Asked if it was important that Australia voted 'yes', he said: "I can't say that."
"As Governor-General, I don't have a declared position on this."
Labor spokesman on constitutional recognition, Pat Dodson, lamented the Government's refusal to put the voice to a referendum.
"We're almost back to the start of the process here," Senator Dodson said.
Senator Dodson also urged the Coalition to progress the makarrata commission proposal put forward at the Uluru convention, which would oversee treaty-making a national truth-telling process.
"There's no reason why the Government can't do two things at once," he said.
"You can do these things if there's a will."
