Bakers, florists and other service providers will not be exempt from discrimination laws, meaning they will not be allowed to refuse to provide services for same-sex weddings.

Key points: James Paterson and David Fawcett will not continue the argument that florists and bakers should be able to say no to same-sex weddings

James Paterson and David Fawcett will not continue the argument that florists and bakers should be able to say no to same-sex weddings But they will introduce five other amendments, including having two definitions for marriage

But they will introduce five other amendments, including having two definitions for marriage Labor, Greens have said they will strongly resist changes to parts of the bill around discrimination

The Senate is considering Liberal Dean Smith's same-sex marriage bill and will debate a range of amendments this week.

But the push is over to extend the amendments to include service providers like florists or bakers.

Liberal senators James Paterson and David Fawcett will not continue the argument that those service providers should be able to say no to a same-sex wedding if they are opposed to gay marriage.

But the two senators have flagged they will introduce five other amendments, including having two definitions for marriage.

They want to keep the definition of marriage as being between a man and a woman as well as having a separate definition that covers same-sex couples that says "marriage is the union of two people to the exclusion of all others".

The two senators argue that having both definitions clarifies that both views are legitimately held.

Bid for two definitions of marriage

The amendments also said faith-based organisations should not lose charitable status if they believe that marriage should only be between a man and a woman.

Parents would also be given the right to withdraw children from some classes if they believed what was being taught was in conflict with their moral or religious beliefs.

Senator Smith's bill allows for religious ministers to be able to refuse to conduct same-sex weddings, but senators Paterson and Fawcett want to extend that to civil marriage celebrants.

Attorney-General George Brandis has also flagged a similar amendment.

Greens senator Janet Rice said her party would be resisting any changes that extended discriminations against lesbian, gay, bisexual, transgender or intersex people.

"In particular the bill already contains quite sufficient protections for religious organisations and for ministers who are going to be able to continue to discriminate," Senator Rice said.

"We very strongly believe that civil celebrants should not be able to discriminate," she said, describing it as a secular marriage rather than a religious marriage.

Senator Rice said she had been told about 3 per cent of existing civil celebrants had concerns about conducting same-sex marriages.

"Our position is even those celebrants, as civil celebrants, should have to abide by the laws of the land and so have to solemnise all weddings if approached to do so," she said.

During debate on the bill, Labor senator Helen Polley urged members to consider the "legitimate concerns" about religious freedoms, warning "people will be persecuted" without proper protections.

The Tasmanian senator appeared to back the Paterson and Fawcett amendments but argued the religious protections should be extended even further to cover conscientious objectors.

"We must acknowledge that there may be non-religious objections to same-sex marriage," she said.

"A person's right of conscience should be respected on this issue, even if it's not on religious grounds.

"If someone wishes to publicly or privately object to a change in secular marriage, they should not be persecuted."

Senator Polley is one of the few Labor members strongly opposed to same-sex marriage, but she did not reveal how she would vote on the Liberal amendments, nor on the bill itself.

Tasmanian Liberal senator Eric Abetz told Parliament the Smith bill would change the definition of marriage in a way he says Australians did not consent to.

Senator Abetz was a prominent No campaigner during the same-sex marriage postal survey.

He said the same-sex marriage bill also allowed intersex and people of non-binary gender to marry and that was not what Australians voted on.

"The bill therefore goes further than that which was approved by the Australian people and adopts the ideology of gender fluidity," Senator Abetz said.

'Step down if you want to speak up'

The same-sex marriage debate has provoked unrest within the Federal Government.

One of the strongest supporters of same-sex marriage, Liberal Warren Entsch, has criticised Coalition frontbenchers including Angus Taylor and Zed Seselja, who are demanding extra religious protections for those who do not believe in same-sex marriage.

Mr Entsch said they should not make the argument from the frontbench.

"Do us a favour and step down from your portfolios, you can speak freely then without breaching ministerial codes," Mr Entsch said.

But Senator Seselja called that unreasonable and hypocritical.

He argued they had every right to speak their mind because there is a conscience vote on marriage and there is not a formal government position.