At a meeting on Monday night the council voted in favour of lobbying the federal government and will further raise the subject at the forthcoming Australian Local Government Association national general assembly.
"This is a contentious issue but is one that we should be having discussions around," councillor Helen Burnet said after proposing the motion, as reported by News Corp.
The Aboriginal community will be consulted with regard to deciding on an alternative date, the council voted.
From Canberra, Greens senator Rachel Siewert congratulated the southern capital.
"Well done to the City of Hobart for being brave enough to kick start a conversation about Australia Day," she said.
"A lot of Tasmanians clearly want Australia Day to accessible to everyone, it shouldn't be on a day that some members of the community view as a day of grieving."
Perth's City of Fremantle has previously moved to reschedule some of its Australia Day events and Senator Siewert said others can subsequently see the benefits of becoming more inclusive.
Thousands of people again attended invasion day rallies across Australia on January 26, calling for a change in the timing of Australia's national day.
But not all Hobart councillors were in favour of the full motion, with Lord Mayor Sue Hickey labelling it "headline-grabbing".