It comes as debate again flares over the issue and threatens to cause a divide within the Federal Government.
Same-sex marriage may not be allowed under Australian law, but a major research project shows a solid majority of the public now supports the idea.
The annual Household, Income and Labour Dynamics in Australia study has been tracking the economic and social situation of more than 17,000 people since 2001.
In 2005, around 30 per cent of men in the study favoured marriage equality.
Over a decade, that number has now risen to almost 60 per cent, with young men leading the push for change.
Among women, the proportion in favour has increased to 67 per cent.
The issue is a thorny one for the nation's politicians, with the Federal Government's promise to hold a plebiscite failing to pass the Senate.
Cabinet minister Simon Birmingham is still standing by the Government's position.
"This issue, we've got a clear pathway forward. If the Labor Party just let us give the Australian people a say, people could have their vote, the issue could be dealt with this year, and I suspect we'd have marriage equality by the end of the year."
But the Opposition, maintaining such a public debate would cause unnecessary pain and endanger some Australians, wants a parliamentary vote.
And Liberal backbencher Dean Smith is drafting a private bill on same-sex marriage, renewing tense debate within the governing Coalition.
A number of MPs are backing him, and some say they will cross the floor if a vote is held in parliament.
Tasmanian senator Eric Abetz, who opposes the move, has told the ABC that MPs voting against their own party could threaten the authority of the Government.
"Voting against your Government on a procedural matter which denies the Government the capacity to control the order of business in the House, that is a grave matter. Now, if a piece of legislation were to come up before the House, at the Government's request and behest, then, of course, the right to cross the floor on that particular measure is part and parcel* of what we in the Liberal Party believe in. But to deny the Government the right to be able to control its agenda on the floor of the House is something that would be a very grave matter, indeed."
The controversy is also affecting Prime Minister Malcolm Turnbull's leadership.
Parts of the Coalition are calling for a voluntary, non-binding postal plebiscite to resolve the stalemate.
But Labor frontbencher Penny Wong has dismissed that idea to the ABC, saying it is time the Government dealt with same-sex marriage once and for all.
"We opposed a plebiscite proper, I suppose you'd call it for principle reasons, and those reasons are even stronger, I think, in respect to a postal plebiscite. I think the people concerned inside the Liberal Party are standing up for what they believe in. They've been trying to manage this for some time, and I guess, for many of them, it's become untenable, and Malcolm should respect that."
Labor M-P Tim Hammond has told Sky News the matter needs to be resolved as soon as possible.
"The least harmful, the least expensive and the quickest way just to get this done is with a free vote in the parliament, which could be done as early as next week. Now what's promising about that is that we're seeing signs in a few of the Government backbench ranks that they are coming to the same view."
