Peter Dutton says the Coalition’s policy is to achieve Indigenous recognition without enshrining a voice to parliament in the constitution, seemingly ruling out a key demand by the authors of the Uluru statement from the heart.

The home affairs minister’s comments follow an anonymously sourced story in the Australian newspaper, which said Scott Morrison would “veto” any attempt to insert an Indigenous voice into the constitution.

Dutton said the government was not in favour of a “third chamber” of parliament, a misleading characterisation of the voice to parliament, given that it would not have any veto powers and could not introduce legislation or change it.

Moderate Coalition MPs say they will 'walk with Ken' on Indigenous recognition Read more

Indigenous leaders included the voice as part of the Uluru statement, proposing an advisory board on Indigenous issues and legislation affecting First Nations peoples, which could not be abolished.

An Essential poll published on Friday found a majority of voters were in favour of both recognition and a voice to parliament.

Dutton said the Coalition had been consistent in its views. “We went to the last election with a policy,” he told the Nine Network on Friday.

“We have been clear that we want to work with the people to provide that recognition. We are not in favour of a third chamber or voice.

“We have a strong democracy. We want to see more Indigenous people in the parliament. It is great Ken Wyatt is the first Indigenous affairs minister. He has a process under way. Let him conduct the consultation. Then we will make an announcement about the next step.”

In laying out his roadmap for a reconciliation plan, Wyatt counselled fellow Indigenous leaders to be “pragmatic” about what could be achieved.

He said he would pursue some model of the voice to parliament as set out in the Uluru statement, but said it could be legislated rather than enshrined in the constitution. Wyatt has also reserved the right to call a halt to the process if he does not believe a future referendum would succeed.

Labor, which is working with the government to achieve reconciliation after decades of failures by successive parliaments, has “endorsed and embraced” the Uluru statement in full.

“What the Uluru statement clearly said is that the aspiration was absolutely for an entrenched voice in the constitution,” the shadow Indigenous affairs minister, Linda Burney, told Sky on Thursday.

“And what people are forgetting is that the reason for that is that there would not be the capacity for any particular government with a particular view to be able to get rid of that voice in the way in which we saw Atsic be gotten rid of.”

Atsic, the Aboriginal and Torres Strait Islander Commission, was abolished by the Howard government in 2005.

“What we need to think about,” Burney said, “is on every single social rung – it doesn’t matter whether it’s health outcomes, education outcomes, overcrowding, domestic violence, life expectancy – Aboriginal people and Torres Strait Islanders are on the bottom rung.

“And that is because there has not been Aboriginal people at the table helping design the legislation and advise on legislation and programs. And that’s what this voice is actually about.”

Greens senator Rachel Siewert said it was “disheartening” and “deeply disappointing” some members of the Coalition were not listening to Indigenous Australians.

“First Nations peoples have undertaken an extensive and exhaustive process of consultation to develop and present the Uluru statement from the heart. It is clear that they want a Voice to parliament enshrined in the constitution. There are other examples overseas of similar approaches that are operating successfully,” she said.

“We are urging the government to outline a clear timetable for implementation of the whole Uluru statement from the heart. This includes a process for developing a national treaty and regional and local treaties. “To be a truly reconciled nation and close the gap, we must listen to First Nations peoples and take meaningful actions, actions that they have eloquently outlined in the Uluru statement.”

Splits have already emerged in the Coalition party room over what should be included in any referendum, despite the process barely progressing beyond a speech and a pledge to move towards a solution.

The voice to parliament’s characterisation as a third chamber began almost as soon as the Uluru statement was handed to the then prime minister, Malcolm Turnbull. It has remained as one of the biggest hurdles to moving the process forward since.

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“There is no suggestion of a third chamber,” the Labor leader, Anthony Albanese, told the Nine Network. “This is spin in order to argue against the Uluru statement from the heart.

“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 where issues affect First Nations people, Aboriginal and Torres Strait Islanders, they should be consulted.

“It is as simple as that. It is about respect. We need to move this issue forward in terms of constitutional recognition in a way consistent with what Indigenous Australians are saying they want.”

Greens senator Rachel Siewert said it was “disheartening” and “deeply disappointing” some members of the Coalition were apparently not listening to Indigenous Australians.

“First Nations peoples have undertaken an extensive and exhaustive process of consultation to develop and present the Uluru Statement from the Heart. It is clear that they want a voice to parliament enshrined in the constitution. There are other examples overseas of similar approaches that are operating successfully,” she said.

