Among the proposed changes are allowing people to be able to register to vote on the day, rather than the rolls being closed off weeks before.
GetUp chief executive Larissa Baldwin-Roberts said the suggested amendments would undo years of suppression and ensure thousands of Indigenous people could vote.
"We saw how similar reforms in the Northern Territory elections increased voter turnout," she said.
The Greens have called for similar reforms, along with options to allow more remote polling and to make voting by phone a possibility, similar to what took place at the last federal election.
The Indigenous enrolment rate was sitting at 84.5 per cent in December, up 2.6 per cent since June.
However, there were still 87,000 voting-age Indigenous people not enrolled.
By comparison, 97.2 per cent of the total Australian population are enrolled to vote.
The referendum working group is meeting on Thursday, ahead of its final presentation to government on the proposed constitutional change.
The group is examining how the referendum will be worded and any possible amendments.
The national vote will be held between October and December this year.
Liberal frontbencher Simon Birmingham said the opposition was looking for certainty about what the voice would look like.
"I hope we can see a proposal put forward that ensures what Australians are voting on delivers on constitutional recognition, but is constitutionally as conservative as possible to give it the greatest chance of success," he told reporters in Canberra.
"I hope the working group, I hope the government, are pursuing an approach that can give people maximum confidence by having a model that is as careful and conservative in its wording as possible."
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