NACCHO represents over 6,000 ACCHO staff - of which 3,500 areIndigenous - and is the largest employer of Aboriginal and Torres Strait Islanderpeople in Australia.
Less than two days after Indigenous Australians Minister Ken Wyatt was hailed for laying out a pathway to hold a referendum on Indigenous constitutional recognition within three years, a backlash from conservative MPs has forced the government to clarify its approach.
Most also support a treaty with indigenous Australians, the Essential survey confirms.
Any referendum question must be supported by the majority of Australian voters, as well as at least four out of six states.
The voice, which would advise the government on issues relating to the indigenous population, was first proposed in the Uluru Statement from the Heart - a historic 2017 document that was signed off by indigenous leaders.More news: Manchester United renew interest in two players from Serie A
"It never was a third chamber", he told the The Sydney Morning Herald.
Thomas Mayor, who is involved in the union movement, says a voice to parliament would empower Indigenous people.
Labor leader Anthony Albanese on Thursday declared that a successful referendum on recognition was "realistic and doable". "It is as simple as that".
Prime Minister Scott Morrison and his predecessor Malcolm Turnbull have rejected enshrining a "third chamber" of parliament in the constitution.
"A first nation's voice to parliament is a meaningful step towards reconciliation", BHP Chief Executive Officer Andrew Mackenzie said earlier this year.More news: At Vatican, empty tombs add new twist to missing girl mystery
Mr Morrison's constitutional view has not changed but he would consider a legislated national body comprising existing indigenous groups, The Australian reports.
Above all, he is adamant the proposal put to the Australian people must have a strong chance of success.
"When a constitutional (referendum) fails, then it leaves an impact for our people, it will be a significant impact on the psyche".
"Labor has been saying clearly that we would like a voice entrenched in the Australian constitution, and I know that there are many people on the government benches that also believe that", said Linda Burney, the shadow minister for Indigenous Australians.More news: Democrats issue new subpoenas amid investigation into whether Donald Trump obstructed justice