Another pathway could open for legal protections against religious and LGBTQI discrimination directed at staff and students, if the government shares its specific proposals.
Labor has not released draft laws on the two issues, but has handed its legislation to the opposition on a confidential basis.
Prime Minister Anthony Albanese has warned he would not press ahead with the laws if the opposition wasn't on board, saying he wanted to avoid a divisive debate.
"I do not want to see a rancorous debate without achieving an outcome," he told a caucus meeting on Tuesday.
One Labor MP raised concerns about the path towards protecting LGBTQI children at school.
Mr Albanese responded that there was a potential to work with the Greens if the opposition did not support the laws, but that he didn't have the minor party's support on the religious discrimination aspect.
"We are concerned about all the forms of discrimination, if the Greens are willing to support the rights of people to practice their faith, then that would be a way forward but we don't currently have that," he said.
While the Greens and cross bench have offered to work with the government and even engaged in discussions together, Labor has refused to share its legislation.
"It is deeply frustrating to have theoretical discussions with Labor when we could be having very practical discussions about how this law works, and how we can make it happen," Greens senator David Shoebridge told reporters.
"We have one set of interests and only one set of interests ... It's about protecting queer and trans kids from being discriminated against at school."
Ahead of her valedictory speech, Greens senator Janet Rice added she would continue "fighting outside parliament against any bill that acts only as a Trojan horse for hate".
Mr Albanese gave examples of discrimination he didn't want to see, including a woman in Bankstown having a hijab removed, a child at a Jewish school being harassed or a student being discriminated against because of who they are.
The Australian Law Reform Commission has recommended scrapping laws that allow religious schools to discriminate against staff and students on the basis of their faith.
LGBTQI advocates have called for the scrapping of discrimination carve-outs for religious schools, but faith groups argued their communities and values should be protected, and that people have options when it comes to choosing schools.
The opposition has remained silent on a final position, but has called for the government to bring on the bill so it can be debated in public.
But coalition members have expressed in-principle support for religious schools being able to preference and hire people in line with their faith.
Religious schools should be allowed to hire workers who reflect their faith but shouldn't then be able to sack staff based on their gender identity or sexual orientation, Workplace Relations Minister Tony Burke said.
Australian Associated Press
