Throsby MP Stephen Jones' same-sex marriage bill has proven less popular than a rival Greens bill as the Federal Government wraps up the first stage of its inquiry into the issue.
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A Senate committee into same-sex marriage finished the consultation phase of its inquiry last week.
An online survey undertaken as part of the consultation process found two-thirds of all respondents believed the law should be changed to allow gay couples to marry.
Of those, 64 per cent expressed support for the Greens bill, put forward by lower house MP Adam Bandt.
In a separate question put to the same group, 60 per cent supported Mr Jones' bill.
Mr Jones wants to amend the Marriage Act to enable same-sex marriages to be recognised but places no obligation on churches to solemnise the unions.
Mr Bandt's more far-ranging bill seeks to allow marriage regardless of sex, sexual orientation or gender identity.
The two bills are making their way through Parliament, but may change depending on the final recommendations of the Senate committee due to hand down its report on June 18.
A spokeswoman said Mr Jones was satisfied with the content of his bill and not worried about its popularity against that of Mr Bandt's.
"Their intended outcomes are effectively the same and we need to focus on the issue rather than the politics," she said.
Mr Jones has previously accused the Greens of grandstanding after the party put forward its rival same-sex marriage bill in January.
At the time he said he did not rule out negotiating with the Greens, but said he was focused on trying to draft a bill which maximised the support from Labor and Coalition ranks.
