Bill Shorten says the party is willing to work with the Coalition to make sure no child is denied human dignity.

Bill Shorten has offered Scott Morrison Labor’s support to repeal discrimination law exemptions that allow religious schools to turn away and expel gay students.

Shorten made the offer at a media conference on Friday, after Morrison reversed his position by saying schools should not be able to expel students based on their sexuality.

The proposal for a bipartisan approach is a significant shift for Labor, and came just hours after its deputy leader and education spokeswoman, Tanya Plibersek, reiterated the party’s policy that it had no plans to roll back existing discrimination law exemptions.

Morrison has expressed in-principle opposition to discrimination, while backing a recommendation in the Ruddock review to enshrine religious schools’ power to discriminate on sexuality. Labor’s offer stops short of committing its MPs to vote for a Greens bill to repeal religious exemptions.

On Thursday, Morrison said he was “not comfortable” with religious schools turning away gay students, and that students should not “be kicked out because they have a different sexuality to what might be believed to be the appropriate thing by a particular religious group”.

Ruddock religious freedom review: what is it and what do we know so far? Read more

Shorten told reporters in Melbourne that “there is no case to discriminate against any kid based on sexuality”.

“Those laws are out of date, if they were ever in date.”

Shorten said he had spoken to school administrators and “they don’t even use the existing laws”.

“So if Mr Morrison wants to back up his statement, I will work with him to make sure that no child is denied human dignity.”

The offer is limited to children, meaning Shorten has made no commitment to reconsider exemptions that allow religious schools to fire gay teachers in every state and territory except Tasmania.

Shorten’s office released a letter to the prime minister on Friday afternoon in which he laid out the offer, and also called on the government to release the full Ruddock review, saying both “Labor and the Australian people are being kept in the dark.”

Earlier, Plibersek said it was “wrong and irresponsible” for schools to turn away gay students and “frankly abhorrent to send that message to a child that something is wrong with them”.

Plibersek told reporters in Sydney the “vast majority” of schools did not use their exemptions to the discrimination law to sack gay teachers or expel gay students, but confirmed it was “not Labor’s plan to reduce any of the existing exemptions”.

The Ruddock religious freedom review recommended that the federal Sex Discrimination Act be amended to provide “that religious schools may discriminate in relation to students on the basis of sexual orientation, gender identity or relationship status”.

The federal law already contains exemptions for religious schools that allow discrimination, although the states, including Queensland and Tasmania, prohibit discrimination against students based on their sexuality.

The Ruddock review proposed adding several safeguards – discrimination must be “founded in the precepts of the religion”, a school would have to publish its policy on discrimination and it must have “regard to the best interests of the child as the primary consideration in its conduct”.

On Thursday, Morrison told 3AW radio that “religious schools should be able to run their schools based on their religious principles” and claimed that “no one” was calling for existing exemptions to be repealed. Later that day, he he clarified that he believed schools should not be able to expel students based on their sexuality.

The right to expel gay children from school isn't about freedom; it's about cruelty | David Marr Read more

The prime minister said the Ruddock review was focused on “what is necessary to ensure a positive right of freedom of religion but at the same time not to disrespect or disenfranchise the rights of other Australians” including on the grounds of sexuality.

The Greens education spokeswoman, Mehreen Faruqi, noted Morrison’s backdown and called on him to “do something about it” by supporting the Greens bill to remove anti-discrimination exemptions for religious schools.

Mehreen Faruqi (@MehreenFaruqi) Well do something about it.

Vote for the @Greens bill to remove anti-discrimination exemptions for religious schools.https://t.co/YFPGsEGIBJ

Phillip Ruddock has previously told Guardian Australia that his religious freedom review panel found that “in large measure” the issue of firing teachers and expelling students “was not a problem”.

However, he said some religious schools felt that “their ethos, part of their belief system” could be “unreasonably challenged”, for example if students proselytised against religious teachings.

Alex Greenwich, the independent state MP for Sydney and former co-chair of the yes campaign for marriage equality, said the Ruddock review had recommended “legislating bullying”, calling the proposal “a disgrace”.