Conservative Prime Minister Malcolm Turnbull's policy is for a national vote, or plebiscite, but that proposal was defeated in the Senate in late 2016.
Turnbull has said the issue will not be revisited until after the next election due in 2019.
Same-sex marriage is supported by 61 percent of Australians, according to a Gallup opinion poll in 2016.
"I think there is momentum in relation to having the matter resolved one way or the other," Immigration and Home Affairs Minister Peter Dutton told Sky News television.
"I'm working on the assumption that this issue is going to be dealt with in this parliament," Dutton said in proposing a postal vote.
Opponents to a national vote argue it would be divisive and that a parliamentary vote could easily decide the issue.
There have only been three plebiscites in Australian history, two relating to conscription during World War I, and one to choose a national song in 1977.
Dutton said a postal vote would not require legislation to establish, would not be compulsory and would be a more cheaply administered option than a regular plebiscite.
Opposition Labor Party leader Bill Shorten took to social media today to slam Dutton's proposal.
"The postal plebiscite is a policy for a government that has neither the intellect to know what to do, nor the courage to do what is right," he tweeted.
Failure to move forward on the matter is seen as a blow to Turnbull's popularity and has given rise to frustrations that he has failed to live up to his progressive reputation.
Earlier in July a government senator said he was drafting a private members bill aimed at legalising same-sex marriage, but Turnbull said he would not allow the bill to be voted on.
(Reporting by Josephn Hinchliffe; Editing by Michael Perry)