Opposition Leader Peter Dutton confirmed the Liberals would back constitutional recognition for Indigenous people, but would not support a voice to parliament and executive government.
But the prime minister said Mr Dutton had been part of a government that was in power for nearly a decade but had not advanced the issues they now called for.
"We've waited 122 years to recognise in our constitution the privilege that we have of sharing this continent with the oldest continuous culture on earth (and) I say to Australians: do not miss this opportunity," he told reporters in Sydney.
"We need to acknowledge that with the best of intentions and good will, what we have done until now is not working.
"We need to consult on matters that affect Aboriginal and Torres Strait Islander people."
Mr Dutton said he had approached the proposal with an open mind and disputed claims by the prime minister that he had been consulted on the terms of the voice.
"The prime minister misrepresents those meetings ... I wouldn't frame it as a genuine engagement," he told ABC Radio.
Asked if his position could ultimately put him on the wrong side of history, Mr Dutton said he was in favour of practical outcomes for Indigenous people.
"The voice has turned into ... an opportunity for there to be input into every aspect of government work," he said.
"I don't believe that that is going to deliver the practical outcomes to Indigenous Australians that we all crave."
But Mr Albanese said the opposition leader's repeated referral to a "Canberra voice" demonstrated the disingenuous nature of the Liberal Party's position.
"Peter Dutton underestimates the goodwill and the generosity of so many Liberal and National Party voters who will ... support a voice and support constitutional recognition," he said.
"This is about whether we as a country can be optimistic ... can come to terms with the fullness and richness of our history, can express our pride in sharing this continent with the oldest continuous culture on Earth, or whether we shrink in on ourselves.
"This is a divisive position that Peter Dutton has taken, this is opportunistic."
Indigenous academic and one of the Uluru Statement's architects Noel Pearson said it was a sad day for Australia that there would not be bipartisan support for such an important national enterprise.
"I've got a great belief that the Liberal Party is greatly out of step with the sentiment of the Australian people on this issue," he told ABC Radio National.
Yet Liberal senator Simon Birmingham insisted there was still an opportunity for bipartisanship and said the party supported constitutional recognition.
"There can still be a means of salvaging something that can provide for the country a unifying, bipartisan moment," he told ABC News.
"Something that is achieved without the type of risks or concerns to constitution and operation of government that have increasingly been discussed through the course of this debate."
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