Speaking on ABC's Q&A, Ms Reid said she was part of discussions which, in May, reached a unanimous position calling for a constitutionally entrenched Indigenous voice to Parliament.
Prime Minister Malcolm Turnbull rejected the proposal in October, claiming it was not "desirable or capable of winning acceptance".
"Why won't you respect our proposal to take it to a referendum and put it to the people like you put the equality, marriage equality to the people? Because polls reveal up to 61 per cent of Australians are in support of this proposal," Ms Reid asked the Prime Minister, who was the sole panellist on last night's program.
Mr Turnbull re-asserted his position that the advisory body would, in effect, be a third chamber of parliament.
"That would mean that that assembly would have the right, if it chose, to examine every piece of legislation," he said.
"As to its prospects at a referendum, let me tell you honestly, as someone who has had some experience in how easy it is to change the Australian Constitution, it would have no prospect of success whatsoever."
Ms Reid responded by calling for bolder leadership.
"I think the Prime Minister continues to undermine our democracy," she said.
"If that's his position, then we clearly need a leader with some courage.
"In his dismissal of the Uluru statement, [the Prime Minister] clearly showed no respect to the Aboriginal and Torres Strait Islander people who participated in those significant dialogues."
Mr Turnbull pointed to Aboriginal politicians such as Ken Wyatt and Linda Burney as evidence of existing Indigenous voices in parliament, but Ms Reid argued they both had their own "political positions".
Asked about the next steps for Indigenous recognition, the Prime Minister said he was working to set up a parliamentary committee to explore the options.
"What we are doing... is a joint committee of the parliament to consider all of the recommendations that have been made on the subject of recognition of First Australians," Mr Turnbull said.
"And to consider them all, and then present... the options that we can pursue to go forward."
Mr Turnbull said he was working with the federal opposition to determine the committee's terms of reference.