A second private bill, containing more protections for religious protections and conscientious objection, is being pushed by conservative coalition MPs, many of whom campaigned for a "no" vote.
Almost eight in 10 Australians who received the Australian Bureau of Statistics same-sex marriage survey form have already had their say.
Liberal senator James Paterson is to introduce bill that would override any anti-discrimination laws passed by state and territory level legislatures.
A rival bill from Liberal senator James Patterson was withdrawn, with the Victorian saying a majority of senators believe the Smith bill "is where we should start".
Any changes the lower house makes to the bill would need the approval of the Senate.
The motion details that Thursday's debate on the bill "shall have precedence over all general business until not later than 6pm". While the freedom to have religious beliefs is also protected unconditionally, the manifestation or expression of those beliefs or religion may be subject to limitation where it impacts upon other fundamental rights'.
The debate will continue the week of 27 November, with amendments reserved for that week.
"There won't be a government position, there won't be a party position", Senator Cormann said.
Liberal senators Linda Reynolds and Jane Hume, Labor's Penny Wong and Louise Pratt, the Greens' Richard Di Natale and Janet Rice, Skye Kakoschke-Moore from the Nick Xenophon Team and Derryn Hinch all signed the motion with Smith.
"I will now work constructively with my parliamentary colleagues over the coming weeks on amendments to ensure that the strongest possible protections for the freedoms of all Australians are enshrined in the final legislation", he said.
In a further rebuke, he said while parliamentarians would be entitled to put forward and debate amendments, he believed ideas such as Senator Paterson's "would have virtually no prospect of getting through the Parliament".
Education Minister Simon Birmingham also voiced concerns.
Senator Smith's bill, which has cross-party support, contains provisions so that ministers of religion could not be forced to conduct same-sex marriages.
Earlier, at a rally in Melbourne, Shorten promised: "Today we celebrate, tomorrow we legislate".
'That's the goal, ' Prime Minister Malcolm Turnbull vowed ahead of the Australian Bureau of Statistics announcing on Tuesday which way more than 12 million people voted on the issue.
Western Australian Senator Dean Smith is expected to introduce his Marriage Amendment (Definition and Religious Freedoms) Bill 2017 to Parliament later Wednesday.
The Human Rights Law Centre said Paterson's bill was "a Trojan horse to allow unprecedented discrimination".