Until now, it looked like the parliament would have to choose between two competing bills from two Liberal senators: the moderate-backed Dean Smith bill and the James Paterson bill, favoured by conservatives who wanted stronger protections for religious objectors in the bill to legalise same-sex marriage.
Senator Brandis added that the amendment was not, nor was there, a government position on the matter.
If the official count of the same-sex marriage survey matches myriad opinion polls, same-sex Australian couples will be able to marry by Christmas.
Ultimately it was up to parliament to decide which bill was the "vehicle to facilitate" the parliamentary debate.
The attorney general, George Brandis, has said he will move a new amendment to allow civil celebrants to reject same-sex weddings as the Coalition gears up for a legislative fight about extending religious protections beyond the cross-party bill.
"If we change the law to allow same-sex couples to marry, and we don't at the same time make provisions for religious freedom, then the. freedom of conscience and freedom of speech of some people could suffer", the Victorian Senator said on Monday.
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.
Cabinet minister Mathias Cormann said his personal instinct was the Smith bill was "probably a good starting position" because it had been subject to a cross-party Senate committee process.
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If the vote is "no", the motion will be dismissed and Smith will table his bill. Liberal senator James Paterson's bill is also far more stringent.
Some 22 bills seeking same-sex marriage have been introduced into the federal parliament since 2004.
"If there is a Yes result on Wednesday, Australians will have voted for true equality for all Australians - not an unfettered right to discriminate for people who voted No", said HRLC director of legal advocacy, Anna Brown.
President, Fiona McLeod SC, said while the Paterson Bill acknowledges concerns of those holding traditional views of marriage, 'it goes well beyond the issue of marriage in a number of crucial respects'. This Bill will encroach on many of these protections in an extraordinary and perilous way'.
The Human Rights Law Centre said Paterson's bill was "a Trojan horse to allow unprecedented discrimination".
Nevertheless, discussions are well under underway as to how same-sex marriage might be legislated.
The senator will instead attempt to negotiate amendments to the bill favoured by both Prime Minister Malcolm Turnbull and Labor leader Bill Shorten, which has already been introduced to the Senate.