Hobart City Council has voted in support of gay marriage equality on social inclusion grounds.

The council will now write to the Federal Government expressing its support for a conscience vote on the issue.

Lord Mayor Sue Hickey said aldermen had been lobbied by both sides of the debate and wanted the Federal Government to prioritise debating changes to the Marriage Act.

"The council decided that it was core business because it's about social inclusion," she said.

"I think the view of the council was that society is changing and, at some level of government, a little bit of courage has got to be taken to stand up and reflect the changing social norms.

"We can't change the law, it has to go to the Federal Parliament but what we are saying is that, as representatives of the community, the Hobart City Council decided to support this motion.

"We've got to show a bit of leadership and say, 'well any loving God would not discriminate on this basis. You certainly can't discriminate by any other level by law'."

Chairwoman of Rainbow Communities Tas Inc, Deidre Murray, spoke to Monday night's council meeting and said the vote was "great support at ground level".

"It's very pleasing to see that the local council does realise that there is a lot of gay families and a lot of people who are concerned about the future of our children," Ms Murray said.

"From my point of view, as a mother of two young children, it's really important that my children grow up with tolerance and understanding, and that they be accepted as they are, as our family is.

"My children go to school thinking that having two mums is normal, and I expect their normal to be valued and respected as every other child's is.

"I think it's really important that we endorse same-sex [marriage] or marriage equality so children have that for the next generation.

In March, the Launceston City Council voted down a motion supporting marriage equality.

The motion was lost eight votes to three, with one alderman abstaining.