Burney, whose Labor party supports the referendum, said she expected a "pretty ugly" no campaign run against the proposed change, but believes the "Australian public, in the main, is ready for constitutional reform".
Indigenous communities are the most disadvantaged in Australia.
Indigenous Australians Minister Ken Wyatt has committed to holding a referendum on constitutional recognition within the next three years.
Two years ago, more than 250 indigenous leaders held a summit not only calling for recognition, but also for an advisory role to parliament - something that activists have been calling for since the creation of the constitution in 1901.
"We're not in favour of a third chamber or a separate voice", senior minister Peter Dutton told the Nine Network on Friday.
"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 the national vote, Mr Wyatt sought to calm his colleagues.
It was immediately rejected by then-Prime Minister Malcolm Turnbull, who described it as a "third chamber" - a claim Indigenous advocates have repeatedly rejected.
Labor leader Anthony Albanese on Thursday declared that a successful referendum on recognition was "realistic and doable".
Wyatt told News Corp on Thursday that indigenous people could have "multiple voices" to parliament.
"It is not a third chamber".
Aboriginal and Torres Strait Islanders lived on the land for tens of thousands of years prior to British colonization. "It is as simple as that".
Australians must return to the polls by 2022 after Prime Minister Scott Morrison's conservative coalition government was returned in a "miracle" election win in May.
Mr Morrison's constitutional view has not changed but he would consider a legislated national body comprising existing indigenous groups, The Australian reports. As with indigenous people around the world, in Australia they face shorter life spans than other residents and greater chances of disease, homelessness and incarceration.
"Sometimes we can aspire to an optimum outcome but we also have to accept that there is a pragmatic element to (a) constitutional referendum and I would rather us in the psyche of this nation have a win on a referendum than to have a loss".