Faith-based schools could still have the right to discriminate against gay and transgender students into 2019 after the coalition and Labor failed to reach a deal on law changes.
But Labor Senator Kristina Keneally says her party is still open to negotiations with the government, as the final federal parliament sitting week for the year draws to a close.
"We have another day of parliament," Senator Keneally told ABC Radio on Wednesday evening.
"If the prime minister and the attorney-general are willing to have a constructive conversation, our door is always open to that."
Even though the government and opposition both agree the largely redundant legal right for religious schools to discriminate should be removed, they have not been able to agree on the wording of legislation.
Labor objects to the government's inclusion of a clause allowing schools to teach in accordance with their religious beliefs, believing it broadens the ground for discrimination.
But the government says the bill "looks after kids", while preserving religious freedom.
There is no evidence gay children are discriminated against by religious schools which have indicated they don't use the right to do so.
But the issue arose following a leaked religious freedom review report which raised fears about potential discrimination and Prime Minister Scott Morrison vowed to have it resolved by the end of the year.
The prime minister on Wednesday called on Labor to support his bill aimed at dealing with the issue, and said he was willing to have all MPs vote according to their conscience.
But Labor deputy leader Tanya Plibersek said the government had complicated every effort from the opposition for progress, just as it did with marriage equality.
