In a narrow vote at the annual National General Assembly of the Australian Local Government Association (ALGA), delegates approved a motion for councils to consider ways to lobby the federal government to switch the date from January 26.
Only 126 delegates from the 560 councils represented at the meeting in Canberra voted on the motion, which was passed by the slimmest of margins with 64 in favour and 62 against.
Hobart City Lord Mayor Sue Hickey, who led the push for change, says Australia Day shouldn't be on a date that many Indigenous people regard as "Invasion Day" because it marks the arrival of the first fleet from England in 1788.
But her big city counterparts in Melbourne and Brisbane oppose any change.
"How ridiculous," Melbourne City's Lord Mayor Robert Doyle told 3AW on Wednesday.
"It's a wonderful celebration of our nation and I think these sorts of calls are just a waste of time."
Brisbane City Council voted against the motion after a heated 45-minute debate on Tuesday.
"It will not support any further moves to change the date," a spokesman said.
Huge protests by Indigenous Australians and their supporters were held on Australia Day this year amid growing calls to change the date.
The City of Fremantle in Perth also moved some of its Australia Day events to January 28, citing cultural sensitivities.
Ms Hickey said she feared Australia Day would continue to be marred by protests until the date was switched.
"There are 365 days of the year, so why not move it to one that avoids all the protests?," she told AAP.
"This shouldn't be an emotional debate, it should be a rational debate."
Ms Hickey said she didn't have a specific date in mind for when Australia Day should be celebrated, but believed it should be a long weekend and possibly at the end of January.
"If we can spend a weekend celebrating the monarch of England's birthday I can't see why we can't be celebrating all that's fantastic about being Australian," she said.
The push for change has been backed by the City of Sydney's Lord Mayor Clover Moore, who oversees the massive Australia Day celebrations on Sydney Harbour.
"The lord mayor of Sydney supports the campaign to change the date and will discuss the issue with the City's Aboriginal and Torres Strait Islander Advisory Panel in the near future," a spokesperson said.
The ALGA Board will meet in July to consider what action to take. However, there is likely to be stiff opposition in federal political circles to any date change.
Assistant Minister for Immigration and Border Protection Alex Hawke said most people support Australia Day on January 26.
"This sort of top-down, elitist conversation will not build a national consensus on important questions such as our national day," he said.
Prime Minister Malcolm Turnbull has previously said the government does not support a change.