The Senate will debate discrimination protections for LGBT teachers on Wednesday and Thursday after a motion moved jointly by Richard Di Natale and Penny Wong was accepted without a vote.

The tactic to force debate on the Greens private senator’s bill means the issue will be considered before the Wentworth byelection on Saturday, despite Scott Morrison refusing to give bipartisan support for ending religious exemptions to discrimination law for teachers and school staff.

The move will add pressure on Morrison, who already faces a divided party. The deputy Liberal leader, Josh Frydenberg, Wentworth candidate Dave Sharma and Liberal senator Dean Smith – the architect of the successful marriage equality legislation – have called to protect teachers.

Labor and crossbench senators including Derryn Hinch, Tim Storer and Centre Alliance support ending the exemptions for religious schools and the Greens corralled them to support a suspension of standing orders after question time on Wednesday.

Instead, a motion to debate the bill for one and a half hours on each of Wednesday and Thursday passed after question time with general agreement, and without the Coalition forcing the Senate to vote or to suspend standing orders.

No vote on the substance of the bill is expected this week, meaning the private senator’s bill cannot pass the Senate until the 12 November sitting week and then faces an uncertain future in the House of Representatives.

The Greens’ bill amends the Sex Discrimination Act, removing the current exemptions for religious educational institutions to discriminate against students and teachers based on sexual orientation, gender identification, marital status or pregnancy.

The Greens’ LGBTIQ+ spokeswoman, Janet Rice, told Guardian Australia that Labor and the Liberals must support the bill “no ifs or buts”.

“The people of Wentworth overwhelmingly voted for equality and against discrimination last year,” she said.

Di Natale said the Senate debate was “an opportunity to get the government and the Labor party on the record” about ending discrimination before “the pressure comes off” after the Wentworth byelection on Saturday.

“We’ve seen some shocking policies in the lead-up to Wentworth,” he said. “What we are trying to do here is make sure that they can’t just wriggle off that hook.”

On Wednesday a group of 47 organisations including the Human Rights Law Centre, Amnesty International, Australian Marriage Equality, the Equality Campaign and Just Equal called on the government to “amend outdated anti-discrimination laws to ensure all LGBTI people are treated with fairness and equality”.

Debate was sparked last week by the leak of recommendations from the Ruddock religious freedom review, including to amend the federal Sex Discrimination Act to provide that religious schools may discriminate in relation to staff and students on the basis of sexual orientation, gender identity or relationship status, albeit with safeguards.

In submissions to the Ruddock religious freedom review, many religious organisations, including the Catholic church, the Anglican archdiocese of Sydney, Christian Schools Australia and the Freedom for Faith group called for a religious freedom act to give religious institutions a right to uphold their values in employment practices.

Backlash prompted first Labor and then the Coalition to promise to end exemptions that allow discrimination against students on gender and sexuality grounds.

But when Bill Shorten offered bipartisan support to end exemptions for school staff, Morrison drew the line, refusing to back Frydenberg’s public calls for teachers to also be protected.

In question time on Monday and Tuesday, Morrison called on Labor to “act in the area of absolute consensus” on students but said that the issue of teachers – like “many other issues” – should be dealt with after the release the Ruddock report.

He also refused to release the Ruddock report, citing the fact its recommendations were already made public by leaks to the media.

Ending religious discrimination law exemptions has been Greens policy since before the 2016 election.