Forty-two thousand years after their deaths, the remains of Australia's oldest known people â€" Mungo Man and Mungo Ladyâ€"are sitting in safes as debate rages on what to do with them and who has the power to make the decisions, Aboriginal groups said.

Mungo Man's repatriation to his traditional lands in the Willandra Lakes area in NSW from storage at the Australian National University in November made international headlines.

But the National Congress of Australia's First Peoples said this week a permanent keeping place now needed to be provided for them and the nation should consider a monument to mark their importance.

Mungo Man's discovery in a dry lake bed in Mungo National Park, about 750kms west of Sydney, in 1974 was hailed as a major archaeological discovery.

For 40 years the remains of the man, who died at about the age of 50 and was buried on his back with his hands crossed in his lap and covered in red ochre, was kept at the ANU in Canberra.

Mungo Lady was found in 1968, also around the now dry Lake Mungo, in 1968 and had been ritually buried, according to the National Museum of Australia.

"I think the fundamental thing is about a national keeping place that has the proper temperature control facility or a statue onsite, maybe out there where the remains were uncovered," Congress co-Chair Rod Little said.

"And perhaps a national monument or a place within the national capital or NSW, but recognition of Mungo Lady and Mungo Man."

Mr Little said they should be embraced by the nation.

"They should be taken care of by the whole of this country," he said. "The government can do it.

"If the government can spend a hundred million dollars on a war memorial in France or somewhere like that then surely they could spend the same amount of money on a keeping place for First Peoples because we are a part of this continent's history and that needs to be recognised and valued."

An Aboriginal advisory group made up of members from three traditional owner groups of the Willandra Lakes Regional World Heritage Area â€" the Mutthi Mutthi, Ngiyampaa and Paakantji â€" is currently tasked with providing advice on managing the area.

Another group related to the Paakantji, the Parintyi, is not represented on the advisory group but is seeking to have the lands of Mungo Man and Lady handed back to them through the Office of the Registrar of Aboriginal Land Rights in NSW as the rightful traditional owners, Parintyi spokesman Michael Young said.

Mutthi Mutthi representative Patsy Winch, a member of the advisory group, said Mungo Man and Lady were still sitting on shelves in safes.

She said it was disrespectful and against cultural law.

"I would like to see them reburied, like they were supposed to be," she said.

Ms Winch said with so many groups with interests in Mungo Man and Lady from governments to universities and traditional owners it was difficult to tell who was in charge.

"It's all very complicated," she said. "No one knows who the boss is."

Mr Young said they wanted Mungo Man and Mungo Lady kept in storage in a long-term keeping place.

"We're not looking at reburial," he said. "There is no security out there. What we're looking at is a long-term keeping place so we can keep them safe and secure 24/7 rather than them end up on e-Bay as an auction."

Mr Young said they would prefer money to be invested in Aboriginal futures such as training and jobs than in a monument.

"Contemporary Aboriginal ideas have moved on from monuments," he said.

A spokesperson for the Ngiyampaa people could not be immediately contacted for comment.

A spokesperson for NSW's Office of Environment and Heritage said the cultural history of the Willandra Lakes was "complex and requires careful consideration and agreement by many parties".

"We are continuing to work with and consult with the Aboriginal community on appropriate care and future visitor experiences regarding the Mungo story including Mungo Man, Mungo Lady and the Willandra collection," the spokesperson said.

The spokesperson said ancestral remains, including Mungo Lady, were held in secure storage at Mungo National Park at an undisclosed location.

A spokesman for NSW National Parks and Wildlife Service said it planned to begin talks with communities in the Mungo area about handing back the area later this year.