The Resolve Political Monitor poll, conducted for the Sydney Morning Herald and Age newspapers, found 64 per cent approval for the draft working of the voice.
Support was weakest in Queensland (59 per cent) and strongest in Tasmania (73 per cent).
The poll came as an emotive new television advertisement - the History Is Calling campaign - was launched on Monday by the architects of the Uluru statement to encourage Australians to vote for constitutional change.
It is the first TV ad launched by advocates after Prime Minister Anthony Albanese announced a referendum would be held in this term of parliament.
The ad stars Pitjantjatjara and Nyungar man Trevor Jamieson, who tells a group of children sitting around a campfire in the desert how everyday Australians helped First Nations people achieve a voice to parliament.
It depicts conversations between tradies, rugby players, dancers and a family sitting around a kitchen table.
Ms Burney said the poll result was "great news" and praised the advertising campaign as "extraordinary".
"It says that Australians believe in fairness and decency and understand that things have to improve for First Nations people," she told ABC radio on Monday.
The government has proposed adding three sentences to the constitution outlining the creation of a body which would represent Aboriginal and Torres Strait Islander peoples.
It would advise parliament and the government on matters relating to Indigenous people but would be "subservient" to legislation.
The government is working with First Nations leaders to determine the timing for a successful referendum and the public information needed for it to pass.
Asked why a voice was needed when Indigenous people could be consulted without it, Ms Burney said it would be a "permanent voice entrenched in the constitution".
"It will be a voice that will advise the parliament, not just government, but the parliament," she said.
"So there will be a point of advice, consultation, negotiation, that people are aware of - it won't be the sort of scattergun approach that sometimes can happen."
She said one of the greatest challenges facing Indigenous people was housing and a voice could have a practical impact on better outcomes in this area.
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