While the proposed bill, sponsored by Liberal senator James Paterson, would legalise same-sex marriage, a number of attached exemptions would give wedding service providers, such as bakers and florists, the right to refuse gay couples.
If supported, Thursday's Senate program will effectively be cleared to prioritise debate on the bill, which will be the subject of a free vote not bound by party positions.
And versions of this marriage equaity bill include so-called religious freedom exemptions that attack LGBT rights, reports Al Jazeera.
The Senator from Victoria said his bill would allow any Australian to declare that they would not participate in a same-sex wedding if it violated their religious beliefs.
"That would be profoundly disrespectful and a rebuke to the people of Australia".
Brady, an Irish citizen, said he saw echoes of the historical discrimination against his own people in the religious freedoms push.
"This is just another attempt to delay passing marriage equality". "I don't think their votes should mean we shouldn't have same-sex marriage, but I do think that their freedoms should be protected and I think my bill is a bill that best does that", he said.
"This moment can't be the moment we literally do the opposite of what the people are saying".
The proposal has already been rejected by Turnbull, who said yesterday while still overseas that "I don't believe Australians would welcome, and certainly the government ... would not countenance making legal, discrimination that is illegal, that is unlawful today".
"You could potentially see a situation where a hire auto company could leave their customers stranded on the way to a marriage ceremony simply because the driver held a thought or belief against it".
When do we get to find out which way Australians voted? "This is even if the belief had nothing to do with religion".
Tasmanian marriage equality advocate Rodney Croome claimed the bill "would effectively allow businesses to hang a "no gays" sign in the window ".
Backed by his conservative colleagues, Senator Paterson released a Private Members Bill on Monday that he argues contains greater religious protections than the one put forward months ago by Liberal colleague Dean Smith.