Liberal senator Dean Smith is expected to get Labor and Greens backing to get his draft laws to allow same-sex marriage onto the Senate floor next week.
The results of the Australian Marriage Law Postal Survey will be released on November 15, a day before the Senate is scheduled to debate what is known as "private senators bills".
Polls are pointing to a victory for the 'yes' vote, after which Prime Minister Malcolm Turnbull has promised the government would facilitate a private bill to change marriage laws.
The Labor caucus resolved on October 17 that Senator Smith's bill "strikes an acceptable compromise" between marriage equality and religious freedoms and it would push for the bill to be passed as quickly as possible.
Greens leader Richard Di Natale is also understood to support the bill being introduced on Thursday, but his party room will finalise its position early next week.
Senator Di Natale told reporters in Canberra on Friday some elements of the Smith bill which "continued to allow discrimination" needed amending.
However, a greater fear was the proposal by conservative senators for an alternative bill.
Senator Smith's bill allows same-sex couples to wed, while creating exemptions for religious organisations that would allow them to refuse to conduct same-sex marriages.
The conservatives want to go further by legislating protections to allow schools to teach students about the traditional view of marriage without having to canvass other views.
"We will wait to see what the final bill looks like before we give a firm commitment as to how we will vote," Senator Di Natale said.
"What we are seeing (from the conservatives) ... is an opportunity to use this as the Trojan horse to further entrench and expand discrimination in law."
Opposition Leader Bill Shorten said Labor would give the Smith bill "very favourable consideration".
"Let's get on with the job of parliament," Mr Shorten told reporters in Perth on Friday.
"This could at least be one moment of us working together."
He said conservatives were "on notice" they would be arguing with the Australian people if they attempted to delay the law change.
The House of Representatives and Senate will sit for two weeks from November 27.