Pat Dodson has accused the prime minister, Malcolm Turnbull, of deliberately throwing out a bipartisan approach to Indigenous constitutional recognition, after a new poll that found 57% of voters — including 38% who typically vote for the Coalition — would support an Indigenous voice model.

The Newspoll, published in the Australian on Tuesday, was the third to find majority support for an Indigenous voice model, which would establish a constitutionally-enshrined representative body of Aboriginal and Torres Strait Islander people to advise parliament on issues affecting Indigenous people.

The poll of 1,632 respondents were asked if they would support the opposition leader Bill Shorten’s pledge to establish a version of the body so Australians could see how it operated before being asked to vote on it in a referendum.

Fifty-seven per cent said they would support it, while 32% were opposed and 11% undecided. Half of all One Nation voters said they would oppose, followed by 48% of Coalition voters, 16% of Labor voters, and 10% of Greens.

Support was strongest among Greens voters (87%), followed by Labor (76%), and the Coalition and One Nation (both 38%).

Dodson, a Yawuru man and Labor senator for Western Australia, said the poll was “encouraging” and showed that Turnbull’s position was not shared by the public.

“Unless the voice of the Australian people can penetrate this government’s ideological opposition to a voice [to parliament for Indigenous people], we are doomed to continue to deal with the First Nations people in this unjust manner,” he said.

He said the Referendum Council had recommended the voice be constitutionally entrenched “because of the past history of governments abolishing any national entity set up under legislation alone”.

In 2005 the Howard government unilaterally disbanded the Aboriginal and Torres Strait Islander Commission (Atsic).

Turnbull rejected the Indigenous voice model, which was a central proposal of the Uluru Statement following 18 months of Indigenous community consultations, in October, claiming it would be a “third chamber of parliament”.

He repeated the criticism last week, saying that having a body attached to parliament that was comprised solely of Indigenous peoples was “inconsistent with a fundamental principle of our democracy, which is that all of our national representative institutions are open to every Australian”.

“If [Shorten] wants to campaign at the next election for there to be a constitutionally entrenched national elected representative assembly able to be voted for and occupied by only Indigenous Australians, he is free to do so,” Turnbull said. “We [the Coalition] believe that all of our national institutions should be open to every single Australian, regardless of their background.”

Dodson said Turnbull’s comments showed “sloppy thinking about race” that “aimed at delegitimising the genuine position of the First Nations as sovereign peoples”.

“First Nations people are not asking for some kind of special treatment,” he said. “We are sovereign peoples and what we are asking for is that parliament do the honourable and just thing, and recognise us for our sovereign status as the First People of this country and acknowledge appalling history which we have been dealt. This is a call for justice, not for one group to be privileged beyond others.”

Prof Megan Davis, a lead author of the Uluru Statement, said Turnbull had “completely missed the point” of a reform that had been approved by constitutional conservatives as “robust and modest”.

“This change is eminently consistent with our democratic tradition and so clearly aimed at improving Aboriginal participation in Australian democracy,” Davis said. “Aussies understood that in 1967 and the Newspoll shows Aussies understand that in 2018.”

A poll of 1,526 people conducted by Omnipoll in August found that 60.7% supported a proposal to “change the constitution to set up a representative Indigenous body to advise the parliament on laws and policies affecting Indigenous people”. That poll also found the proposal had majority support in every state except Tasmania.