Labor has signalled that fixing the law to prevent discrimination against LGBT teachers may be accompanied by changes to allow religious schools to preserve their “ethos” and prevent contradiction of church doctrines.

Despite Labor offering to help the Morrison government legislate to remove discrimination against LGBT teachers and staff, debate on Wednesday revealed some Labor senators have serious reservations that the Greens bill does so without provisions that allow religious schools to preserve their character.

The disagreement could see progress on the Greens bill stall, as the Coalition has indicated it will craft its own legislation with Labor to protect LGBT students from discrimination but does not support the move to protect teachers.

On Monday Bill Shorten wrote to Scott Morrison arguing there was no room for discrimination in religious schools “be it [against] a student or against a teacher”.

Labor supported the Greens in their bid to debate a bill to end the current exemptions for religious educational institutions to discriminate against students and teachers based on sexual orientation, gender identification, marital status or pregnancy.

But on Wednesday Labor senator Jacinta Collins told the Senate it is necessary to “respect the right of religious schools to be run in accordance with their beliefs” and for parents to have their children educated “in accordance with their religious convictions”.

Collins said that schools expect teachers and staff to “respect the ethos, values and principles of the particular faith and not to act in ways that undermine a school’s mission”.

Collins said that while religious exemptions to discrimination law are “out of step with community expectations”, legislators need to ensure that schools are “positively entitled to operate in accordance with their belief and mission”.

She warned that the Greens bill only “addresses one side of the equation” by removing discrimination against LGBT staff but not preserving schools’ ethos.

“We would also like to see in legislation a recognition that religious schools are entitled to require employees to act in their roles in a way that upholds the ethos and values of that faith; and this requirement can be taken into account when a person is first employed and in the course of their employment,” Collins said.

The shadow assistant minister for equality, Louise Pratt, told the Senate that an innate attribute should not be a ground for discrimination but also recognised the rights of parents to “have children educated in accordance with their religious convictions”.

Teachers who behave “totally within the ethos” of a school who “just happen” to be LGBT, unmarried, or pregnant outside marriage should not be discriminated against, she said.

Pratt said there might be “a great deal of conduct [schools] cannot and should not tolerate”, such as teachers engaging in overt conversations about sexual matters or a Scientologist trying to “recruit students at a Catholic school”.

“For some schools, promoting something like marriage equality within the school community might very much be outside the ethos of that particular school,” she said.

Pratt said that an LGBT teacher talking in “an ordinary sense about themselves or their family” and their “status” as an LGBT person should not be grounds for discrimination, because they should not have to hide their identity.

The shadow attorney general, Mark Dreyfus, told Guardian Australia that Labor “has said consistently that we respect the right of religious schools to run their organisations in line with their beliefs and traditions”.

“This is not incompatible with removing discrimination against LGBTI kids and teachers,” he said. “No one should be denied an education, fired or denied employment based on who they are or who they love.”

Dreyfus said Labor will continue to consult with the community and work with “all sides of parliament” to remove discrimination.