As the prime minister revealed at the Garma Festival the possible question that could be asked in a referendum, Ms Burney said there was already a lot of detail available about how the body would function.
She said while the Voice was an important issue, the government would aim to pursue as much consensus as possible about the path to establish it.
"We will not be rushed, and it is very important that this belongs to the Australian people, not to politicians," she told ABC radio on Monday.
"There will be a process, we will not be rushed."
A potential question to be asked in the referendum would be: "Do you support an alteration to the constitution that establishes an Aboriginal and Torres Strait Islander Voice?"
Prime Minister Anthony Albanese said details about the Voice, such as its function and how it operated, would be worked out following consultation but it would only act as an advisory body and not as a third chamber of parliament.
Ms Burney said previous work about the Voice by Aboriginal leaders would not be jettisoned and would be part of the goverment's consideration.
"There is a lot of detail out there in the community," she said.
"If people are going to vote on something, what they need is to have an understanding of why this is important."
While a timeline for the referendum has not been finalised, Labor reportedly prefers holding the vote next year.
Greens First Nations spokeswoman Lidia Thorpe said she welcomed the referendum and wanted all elements of the Uluru Statement to be enacted.
However, she called on the government to implement a treaty with Indigenous people and to follow through on all recommendations from the royal commission into deaths in custody.
"Our priority should be black justice in this country, our priority should be about saving lives today, not waiting for a referendum," she told ABC radio.
Senator Thorpe also led a push in the Senate for the implementation of the United Nations Declaration on the Rights of Indigenous People (UNDRIP) through her private senator's bill.
She earlier made a stand in the Senate while being sworn in, raising her fist - often seen as a symbol of resistance - and branded the Queen a "coloniser" while referring to herself as "sovereign".
Senator Thorpe was then forced to recite the oath of allegiance without the additional words.
Her bill will establish a framework to implement UNDRIP and put in place annual reporting mechanisms.
"UNDRIP is an opportunity for the new Labor government to prove that they are committed to action, not symbolism, for First Nations people," she told the Senate.
"Passing this bill means putting First People in the driver's seat when it comes to making decisions about our communities, our culture and our country."
First Nations Labor senator Pat Dodson said the government agreed with the intent of the bill and should "align our actions with these principles".
"While non-binding, the declaration carries significant moral force, the government supports the aspirational principles underlying the declaration," he said.
First Nations Country Liberal Party senator Jacinta Nampijinpa Price branded the bill "virtue signalling" and called for more practical action.
"This bill is an unnecessary distraction from the important work that needs to be done that we as a coalition have invested heavily in," she said.
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