LGBTQI advocates want the federal government to release draft religious discrimination legislation as a report highlights failings at faith schools.
Labor has given the proposal to the opposition, but is keeping it from the public because it says it wants to avoid a divisive culture war.
Prime Minister Anthony Albanese has said the government would only proceed with bipartisan support.
Equality Australia chief executive Anna Brown said she would like to see a copy of the bill to have a respectful debate.
"We believe strongly that this is an issue that can unite the parliament not divide it and Australians like they did during the marriage equality debate, can come together around the principles of fairness and equality," she said.
In a review, the Australian Law Reform Commission recommended scrapping laws that allow religious schools to discriminate against staff on the basis of their faith.
Workplace Relations Minister Tony Burke said religious schools should be allowed to hire workers who reflect their faith, but not sack employees based on their gender identity or sexual orientation.
"Once you have employed somebody, then all the normal rights against discrimination should be held by that worker," he told ABC radio on Monday.
"It goes to the firing issue. That would be an example of discrimination if it was done for a reason for one of those protected categories."
Shadow attorney-general Michaelia Cash said the government wanted to keep the process behind close doors, labelling the situation absurd and "reaching a farcical level".
"The Albanese government needs to publish its religious discrimination legislation as soon as possible," she said.
James Elliot-Watson, who went to an independent Christian school, said he was discriminated against when he was blocked from student leadership positions because he was gay.
On bipartisanship, Mr Elliot-Watson said politicians should "do their f***ing job".
"It's ridiculous that the prime minister wants bipartisan support when he's in the chair," he said.
"There are young vulnerable people in school right now who will see all this play out and they will feel the fear that I felt, and the shame that I felt, and they need to be protected from that."
A report released by Equality Australia on Monday said LGBTQI discrimination is "endemic" in religious schools and organisations across the country.
It found almost one in 10 of Australia's largest faith-based service providers publicly discriminate against LGBTQI people, while almost four in 10 are silent about their positions on inclusion.
National Catholic Education Commission executive director Jacinta Collins rejected Equality Australia's assertion faith-based schools were seeking to discriminate against staff or students.
She said they were maintaining their religious identity and mission.
"If individuals do not support our ethos, they are able to choose another school for enrolment or employment," Ms Collins said.
"This is what a free, pluralistic society is about."
Labor pledged at the 2022 federal election to update the law to protect religious schools practising their faith, while stopping the discrimination of teachers and students for their sexuality.
Greens leader Adam Bandt said Labor was on the verge of breaking its election commitment and everyone should be able to see the bill.
Mr Burke said it wasn't uncommon for negotiations being discussed in "good faith" to be kept away from the media spotlight and it was reasonable.
"If there's a way of dealing with this constructively with the opposition, then that's a better option for the whole of Australia," he said.
