If the Senate passes the bill by November 30, the House of Representatives could begin debating the bill on December 4.
"There should be strong protections for religious freedoms and freedoms of conscience", Senator Brandis told parliament.
Senator Brandis added that the amendment was not, nor was there, a government position on the matter.
Labor's leader in the senate, Penny Wong, appealed for any proposed amendments to be introduced well in advance so they could be properly considered.
Others to signal support for amendments include the finance minister, Mathias Cormann, and the former prime minister Tony Abbott.
Prime Minister Malcolm Turnbull had already ruled out any chance of the Patterson bill succeeding.
"I believe that the Smith bill will need some improvements", he said.
"If you are a gay man or a gay woman and you go into a florist and say "I'd like to buy a bunch of flowers", it's just wrong and illegal for florist to say "I don't" serve gay people" just as it would be wrong or illegal for the florist to say to an indigenous person "I don't serve indigenous people'".
"I would like to think that the parliament will be able to meet somewhere in between to give effect to the decision of the Australian people ... and to also ensure that we accommodate the legitimate concern of many Australians for there to be appropriate levels of religious protections".
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Senator Smith then rose to introduce his bill.
Earlier, at a rally in Melbourne, Shorten promised: "Today we celebrate, tomorrow we legislate".
"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.
At a press conference Smith suggested a way through could be for an existing parliamentary inquiry into freedom of religion to report after same-sex marriage was legalised, because conservatives are raising issues of parenting laws and free speech that are not relevant to a marriage bill.
"My own view is that people, like ministers of religion, should not be obliged to conduct a ceremony of marriage against the teachings or doctrines of their church and Senator Dean Smith's bill does provide an exemption of that kind", he said.
The draft Marriage Amendment (Definition and Protection of Freedoms) Bill 2017 to legalise same-sex marriage, released by Senator James Paterson yesterday, represents an "extraordinary and perilous" winding back of Australia's anti-discrimination laws under the cover of marriage equality, according to the Law Council of Australia.
"It's clearly a good bill to start with", he said.
Various opinion polls point to a majority "yes" vote, possibly as high as 60 per cent.
One Nation, which controls three votes in the Senate, has reserved its position.