Fairfax Media can reveal that Melbourne Archbishop Denis Hart wrote to Mr Turnbull on Monday to seek a meeting to hammer out a peace deal.

It came as the government agreed to delay introducing its new funding model for at least 12 months for Catholic schools in a bid to stop any of its senators from crossing the floor.

The concession – which has not been officially announced – will not win the support of the Catholic sector but is expected to be enough to placate Liberal Senator Chris Back, who had threatened to vote against the government's changes.

The government was locked in intense negotiations with the Greens and Senate crossbench on Tuesday night, ahead of the expected introduction of its bill into the Senate on Wednesday.

Prime Minister Malcolm Turnbull met with Greens leader Richard Di Natale on Tuesday to discuss a potential breakthrough deal on schools funding.

The Greens are set to finalise their position at a party room meeting on Wednesday.

"Talks are happening at a very senior level but it is still unclear if we can get this through the party room," a Greens source said.

Greens MPs remain concerned about the potential backlash from the well-funded Australian Education Union (AEU) if they strike a deal with the government.

On Tuesday, a former head of the AEU broke ranks with the current union leadership to urge the Senate to support the new funding model as it offers a good deal for public schools.

In his letter Archbishop Hart, who is president of the Australian Catholic Bishops Conference and therefore the most senior Catholic leader in Australia, expressed concern about the new funding model and the amount of money being made available to Catholic schools.

He asked for Mr Turnbull, Education Minister Simon Birmingham and other officials to meet with Catholics bishops to resolve the dispute over school funding.

Archbishop Hart's intervention is significant as it underscores the scale of the disquiet over the Turnbull government's school funding proposal in the Catholic community.

Fairfax Media understands the government will roll over the current funding arrangements for Catholic schools – and other non-government school systems – for a year while a review of the socio-economic status (SES) methodology underpinning the funding model is conducted.

The Catholic sector argues the SES formula – which assesses how much parents can contribute to the cost of their child's education – disadvantages their schools compared with private schools.

Government sources said it would be better to delay the new model for a year than to see it voted down and the current deals remain in place.

But the development has alarmed the private school sector, which warned it could pull its support for the government's school package unless its schools get a reprieve from funding cuts next year.

Greens sources said any special deal to appease the Catholic sector would make it even harder for them to support a deal with the government.

In a letter sent to crossbench senators, former AEU national president Dianne Foggo wrote: "I implore you to support the bill with the tighter funding arrangements raised by the Greens.

"This is one of the most crucial potential improvements for funding for public education in decades – please do something positive to restore equity to our public education systems."

She said she was making a stand even though it would draw "opprobrium" from the union of which she is a life member.

AEU President Correna Haythorpre said the government had become a "complete shambles" and should delay trying to pass its bill.

The process had been "shambolic", she said.

Labor education spokeswoman Tanya Plibersek said Mr Turnbull was more concerned about appeasing his own backbench than creating good policy.

Former prime minister Tony Abbott and former cabinet minister Kevin Andrews raised concerns about the government's changes in a party room meeting on Tuesday but many more MPs spoke in support of the package.