And Treasurer Scott Morrison has also indicated he would also back a new "religious protections" bill, telling Fairfax Media that "right now we should deal with same-sex marriage, but wherever there is a chance to back religious protections, I will do so".

Senator Smith, one of the architects of the same-sex marriage law that Parliament began debating on Thursday, suggested amendments designed to offer legal exemptions for religious groups and organisations may better fit elsewhere in "the legal architecture of our country".

The comments from Mr Dutton and Mr Morrison, two senior conservative figures in the Turnbull government, who both supported a "no" vote, suggest opposition to same-sex marriage is evaporating.

But they also indicate a new debate on religious freedoms could kick off in 2018, following the thumping win by the "yes" campaign in the historic postal survey.

Prime Minister Malcolm Turnbull and leading moderate cabinet ministers including George Brandis and Julie Bishop have already indicated they believe the marriage bill should be dealt with swiftly; Ms Bishop said concerns about religious groups should be addressed in other bills.

Senator Brandis has proposed low-key amendments that would allow civil celebrants to refuse to perform a same-sex marriage ceremony, and which would guarantee people's right to express their religious views of marriage.

Mr Turnbull again emphasised he wanted same-sex marriage legalised by Christmas, which would clear the way for the first weddings to take place in Australia in January 2018.

"There will be a lot of amendments moved, some of them will get up, most of probably won't. But it will be up to members and you'll see Labor members voting different ways on amendments and Liberal members voting different ways on amendments. It'll be Parliament at its best, people will be voting in accordance with their conscience. But believe me, it'll get done," Mr Turnbull said.

Mr Dutton is one of a number of conservatives who have now indicated they will vote "yes" in Parliament for same-sex marriage. He also took a swipe at MPs arguing for amendments that would allow, for example, bakers to refuse to bake a cake for a same-sex wedding.

"I don't support discrimination in any form against any person," Mr Dutton said.

"There are sensible religious protections that are required. I want to make sure that protections are in place as well because I do think the Safe Schools movement will use the debate as a launching pad for their next wave."

"Now that we have that issue resolved, it will give us an ability to have a discussion, I think in the new year, over whether or not there is support for a religious protections bill."

Mr Morrison said he had not yet decided whether he would vote for the same-sex marriage bill and that his position depended on the amendments that were made to the bill.

But in a message to opponents of same-sex marriage, he said: "this issue is settled, the debate is over, but this does not close the debate on other issues such as religious freedoms. It's important that freedoms and protections are advanced".

He has outlined a series of changes, including the ability for parents to withdraw their children from schools if they were taught about same-sex marriage, protections for religious organisations that currently have tax-deductible status or receive public funds, and a "no-detriment" clause for people who believe in traditional marriage.

Mr Smith said that further laws about religious freedoms may belong elsewhere.

"What are the deficiencies in the current laws that exist? How is the Fair Work act currently deficient, how is the Sexual Discrimination Act currently deficient?...what is the legislative remedy and where in our legal architecture does that best fit? It may be the marriage bill, it may not be the marriage bill," he said.

Opposition Leader Bill Shorten said Labor was committed to protecting religious freedom and the ability of churches to practise wedding ceremonies and marriage ceremonies according to their own beliefs.

"I also want to remind Australians that the people who voted 'no' were entitled to their opinion, and I respect that. Having said that, with the legislation, let us get it done," he said.

Labor MP Chris Hayes has also suggested a separate bill that deals with concerns over religious protections could be necessary; colleagues of his such as senator Jacinta Collins have also indicated they could back this approach.

Discussions are already under way behind the scenes between Labor and Liberal opponents and supporters of same-sex marriage, with a second tranche of legislation appearing increasingly likely.