Catholic groups are calling for religious protections to be enhanced as uncertainty clouds discrimination laws.
Labor promised to introduce the changes during this term of parliament ahead of the last election.
But Anthony Albanese told caucus colleagues this week that he would not push ahead without bipartisan support.
His statement seemed to catch the coalition off guard, as their members have not seen Labor's draft laws.
Attorney General Mark Dreyfus is keen to secure cross-party support.
"This is something we've been working on for a long time, we've been consulting about for a long time," he told reporters in Canberra on Wednesday.
"We're very keen to work with the opposition for a bipartisan outcome on this."
Nationals frontbencher Bridget McKenzie signalled one non-negotiable red line.
"At a fundamental level, schools of faith should be able to employ teachers of faith," she told AAP.
Liberal MP Bridget Archer, who crossed the floor to better protect LGBTQI students when similar laws were last debated, called for common ground as a starting point.
"Of course people of faith ought to be protected, that's the starting point that we all agree on," she told reporters.
"If we start from the things that we agree on, and work backwards from there, then consensus opportunities are possible, but you've got to have something to work with."
A lot of harm was inflicted on vulnerable members of the community whenever the topic was discussed, she said.
The changes centre on allowing schools to discriminate against staff and students on the basis of religion.
The government committed to ensuring religious schools couldn't discriminate against students or staff but could continue to preserve religious principles "in good faith" when it came to hiring staff.
Religious schools say both principles can exist, arguing for amended laws that would protect students but allow them to fire teachers who actively undermine their religious theology, not just because of their sexuality or gender.
Christian Schools Australia and Australian Association of Christian Schools said bipartisanship was possible.
"If the religious school has clearly set out their beliefs and conduct expectations, and a staff member deliberately acts to undermine these beliefs, the school should be able to bring the employment relationship to an end," Associated Christian Schools said.
Mark Spencer from Christian Schools Australia responded to fears about gay students being expelled.
"It simply has never happened, won't happen and no one wants that to happen," he said.
An Australian Law Reform Commission review of anti-discrimination laws and religious education institutions will be released on Thursday.
Former prime minister Scott Morrison tried to push his own religious discrimination bill in February 2022.
But five Liberal moderates, including Ms Archer, crossed the floor for an amendment that stripped the right of schools to discriminate against students on the basis of gender identity, sexual orientation or their relationship status.
The laws were shelved and have not been revisited since.
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