The push to change the date of Australia Day has won support from a traditionally conservative quarter.

During a meeting in Canberra, the National General Assembly of Local Government narrowly passed a resolution for the assembly to encourage Australian councils to consider efforts they could take to lobby the Federal Government to change the date of recognition.

After a lengthy debate, the Hobart City Council's motion was supported 64 votes to 62.

Alderman Sue Hickey said she was thrilled at the support for changing the date.

"There is a growing momentum that the 26th of January is not representative of who we are as Australians and it is certainly not inclusive of the first people," she said.

"There were different dates thrown around like, the date of Federation, May 8 which rhymes with mate."

Sue Hickey says it's not a day of celebration for all Australians. ( ABC News: Ros Lehman )

The Australian Local Government Association (ALGA) board will consider the resolution at its meeting next month and decide what action to take, but the push to change the date will not necessarily become ALGA policy.

"It is a very small step on a long journey," Alderman Hickey said.

She said she believed the Federal Government would take an approach from local government seriously.

"Local government is the closest level of government to the people, so I think it is an an excellent avenue to speed up to the Federal Government the concerns of the public," she said.

"I would just like to us to consider a date that is far more considerate for all Australians, so we can stop the protests and move forward."

Some councils have moved their Australia Day celebrations but the Federal Government can dictate when citizenship ceremonies are held.

Primer Minister Malcom Turnbull has previously rejected calls to change the date and a Government spokesman said the position had not changed.

Meanwhile, Assistant Minister for Immigration and Border Protection Alex Hawke said most Australians did not favour a date change and the vote showed local government was "hopelessly divided".

"This close result was at a forum where it is easy for the views of small minorities, such as those of Greens councillors, to be given more importance than those of everyday Australians," he said.

"This sort of top down, elitist conversation will not build a national consensus on important questions such as our national day."