Fairfax Media understands the private school sector is considering withdrawing its support for the government's package because it has not received a similar offer to delay funding cuts for its schools. Education Minister Simon Birmingham has offered concessions to the Catholic school sector Credit:Louise Kennerley During the next 12 months a review of the socio-economic status (SES) methodology underpinning the government's funding model would be 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 to private schools. The concession was expected to be presented to the Coalition party room on Tuesday in a bid to head off a revolt from conservative MPs concerned about a $4.6 billion hit to Catholic schools revealed by Fairfax Media on the weekend.

It meets the demand of WA Liberal Senator Chris Back, who had threatened to cross the floor and vote against the government. Government sources said it would be better to delay the new model for a year for Catholic schools than to see it voted down and the current deals remain in place. Fairfax Media understands Senator Back, Kevin Andrews and Tony Abbott raised concerns in the party room about the school funding model but other MPs such as Julian Leeser, George Christensen and Luke Howarth spoke out in support. Catholic education sector sources said the move, which would deliver them an estimated extra $62 million funding next year, would provide them with welcome "breathing space" but would not convince them to back the government. It came as a former head of the Australian Education Union urged the Greens and Senate crossbenchers to back the government's school funding changes in a dramatic break with the position of the current union leadership.

In a letter to senators, former AEU president Dianne Foggo writes: "I implore you to support a re-engineered Gonski [agreement]. "I agree with Labor that Gonski 2.0 reduces funding that was envisaged in Gonski 1. However, it is on the principle of needs-based funding into the future that the Turnbull government's bills must be supported. "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." The letter was sent to Prime Minister Malcolm Turnbull, Opposition Leader Bill Shorten, senators and current AEU President Correna Haythorpe, who has insisted the new funding model must be blocked entirely.

The Greens are terrified of doing a deal without the backing of the union for fears of the campaign they could unleash against them. Ms Foggo, a life member of the AEU and former Australian Council of Trade Unions vice president, said she expected to receive "opprobrium" from unionists for her intervention. "However I cannot in all conscience stay silent when there is an opportunity to change the current schools funding regime for a better one," she wrote. "Why wouldn't we all support something which will give pre-eminence to public schools in Australia? "Why wouldn't we support a reduction in the over-funding of a small number of private schools?"

Loading The government would need the support of the Greens or 10 members of the Senate crossbench to pass its new school funding model this week. Follow us on Facebook