After rejecting calls to change the date of Australia Day, Prime Minister Scott Morrison says there could instead be another national day to celebrate Indigenous people and culture.

Key points: PM floats the idea of a new national day to recognise Indigenous culture

PM floats the idea of a new national day to recognise Indigenous culture Frontbencher Ken Wyatt, an Aboriginal man from WA, backs idea and suggests holding it during NAIDOC Week

Frontbencher Ken Wyatt, an Aboriginal man from WA, backs idea and suggests holding it during NAIDOC Week Proposal comes as a local council has its powers to hold citizenship ceremonies stripped over moving the date of Australia Day

The PM floated the idea amid renewed debate over national celebrations being held on January 26, the date the First Fleet arrived at Sydney Cove.

Mr Morrison ruled out changing the date, but suggested it would be good to "chat with the Australian people" about the concept of a new national day to recognise Australia's Indigenous history.

"There's a lot to celebrate … and I think we can celebrate the fact that this is the world's oldest living culture," he said.

While not nominating a date for the new day, Mr Morrison noted the ACT now holds a Reconciliation Day public holiday on May 28, which marks the anniversary of the successful 1967 Indigenous referendum.

"We don't have to pull Australia Day down to actually recognise the achievements of Indigenous Australia, the oldest living culture in the world; the two can coexist," Mr Morrison told Channel Seven.

"Australia Day is Australia Day … You can't pretend your history isn't your history.

"That's the day the flag went up in Farm Cove. That's the day the course of the nation changed."

Mr Morrison said he did not want to "get too far ahead of myself on this" and noted that it was up to states and territories to decide on public holidays, but added that he was open to a discussion about another national day.

Rod Little from the National Congress of Australia's First Peoples welcomed discussion of the new national day, but said it needed to be accompanied by "truth telling" about historical injustices.

"There are lots of bad things that have happened in these 200 years [since colonisation] and we're still feeling the effects of that," he said.

Acting Labor leader Tanya Plibersek demanded more details, saying "We can't have a conversation based on a thought bubble in the media".

Event in NAIDOC Week flagged as possible date

Government frontbencher Ken Wyatt, an Aboriginal man from Western Australia, described the idea of an Indigenous day as "a great step forward" and suggested holding the event during NAIDOC Week in July.

"This is something I'd raised previously with a former prime minister," he said.

"NAIDOC Week has been a great week in Australian society."

Mr Wyatt said Australia Day should be left on January 26 because the day was "not about the settlement of this country anymore".

"It is about us as a nation, us as a people, and the melting pot of a society that is working closely to build this nation," he said.

Special Envoy for Indigenous Affairs Tony Abbott withheld judgment on the proposal.

"Because we've already got things like NAIDOC week and national Sorry Day and so on," Mr Abbott told Sydney radio station 2GB.

But he congratulated Mr Morrison for his "attack on those leftie, greenie, guilt-ridden councils that somehow think Australia Day is a day of shame, rather than a day of pride".

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The Federal Government has stripped Byron Shire Council of its right to hold citizenship ceremonies after the local government moved its Australia Day ceremony forward by a day.

Mayor Simon Richardson defended the council's decision to "move a ceremony about 18 hours to reflect history and show some respect to all Australians".

"That particular date causes pain for a section of our community, so it's sort of rubbing salt into their wounds, so why do we have to?"

Three Melbourne councils last year decided to stop holding future celebrations on January 26, and were similarly stripped of their powers.

On Monday, commenting on Twitter about the Byron Shire Council's decision, the Prime Minister insisted January 26 remained a day to reflect on the nation's modern history and said "indulgent self-loathing" would not make Australia any stronger.