Mr Morrison is now calling for a special day to be set aside to recognise Indigenous Australians and their 60,000 years of history.
He told the Seven network he had "personal views" about which date should be chosen but was interested in the views of the community, the states and business groups.
"It would be good to have a chat about it. We should think about it. We don't have to pull Australia Day down to actually recognise the achievements of Indigenous Australians, the oldest living culture in the world," he told Seven's Sunrise.
"The two can coexist."
Every year, the government faces growing calls to change the date of Australia Day out of respect for Indigenous Australians. Many Aboriginal and Torres Strait Islander people feel it is wrong to celebrate a day that marked the beginning of British colonisation in Australia, which led to widespread massacres of Indigenous people and the loss of traditional land ownership.
Ken Wyatt, minister for aged care and the only Indigenous man in the cabinet, said he had raised the suggestion of a new national day to "recognise the oldest continuous living culture" with former prime minister Malcolm Turnbull.
Mr Wyatt said celebrations on Australia Day itself were more about "what we have become" as a nation, rather than a celebration of "settlement" and the First Fleet's arrival at Botany Bay.
Indigenous congress fears 'distraction' in leadup to election
One leading Indigenous group has welcomed the chance for a debate about a new national day, but warned the issue should not become a "distraction" from urgent problems like the over-representation of Indigenous people in prison.
"This is kind of deja vu for us," Rod Little of the National Indigenous of Australia's First Peoples told SBS News, saying these sorts of cultural debates were common in the "leadup to an election".
Mr Little said he agreed with Mr Morrison that January 26 could not be forgotten or erased, but said the day should be marked with a combination of somberness and celebration, more like Anzac Day.
Mr Morrison said that while Australia had a “great national story” it was “not perfect”.
"You don't pretend your birthday was on a different day," the prime minister told reporters in Sydney.
"You look at your whole life experience. Your achievements and a few scars and from some mistakes and things that you could have done better."
Mr Little said the prime minister's comments were "flippant, in terms of the suffering that has been caused by colonisation".
"The effects are still here today," he said.
Deputy Nationals leader's mistake
Deputy Nationals leader Bridget McKenzie told Sky News it was important to commemorate Australia Day on January 26 because it was the day "Captain Cook stepped ashore".
The day actually commemorates the arrival of the First Fleet at Botany Bay, some 18 years after Cook's arrival. James Cook was dead by the time the First Fleet arrived.
Mr Little said a new date could be a chance to educate Australians about the country's colonial history.
"We know we can't undo the history but we can expose the truth of our history," he said.
Morrison blasts 'self-loathing' council
Mr Morrison said some wanted to change the date out of a "deep respect" for Indigenous people, but said the Byron council's decision was a product of "self-loathing".
"Indulgent self-loathing doesn't make Australia stronger," Mr Morrison tweeted on Monday.
"Being honest about the past does."
Byron Mayor Simon Richardson, whose decision sparked the federal blowback, said the celebrations on January 26 caused pain in a section of the community and questioned whether the values of a fair go and mateship were being reflected.
"Is it true mateship to willingly, willfully and continually to celebrate what rightfully is great to be an Australian on a day that some Australians are pained by?" the Greens representative told 3AW on Monday.