Mr Morrison said the funding boost was not a repudiation of the model set up by Malcolm Turnbull and the previous education minister,Simon Birmingham,but:"When there are issues that need to be addressed,we'll address them."

Government documents indicated the new $1.2 billion pot of money will be used for"government priorities for school education",including regional,rural,remote and drought-affected schools.

But federal Education Minister Dan Tehan confirmed individual schools will not apply for money from the fund. Instead,it will be paid to the peak bodies in each state,and distribution"ultimately will be choices that each of the sectors will make"based on their own priorities.

In particular,the Catholic sector has emphasised its desire to keep providing low-fee alternatives to public schools and independent schools,even in wealthier areas where parents could pay more.

"They put a very strong argument that in some cases they do need to be able to offer low-fee offerings because otherwise choice will be taken away from parents,"Mr Tehan said.

He said a"framework"would apply to the $1.2 billion fund,but details were yet to be finalised. But detailed modelling has been done behind the scenes on where the money will likely go.

Sources in the Catholic and independent sectors told Fairfax Media about $718 million would flow to Catholic schools nationally,while about $485 million will go to independent schools.

Labor's education spokeswoman Tanya Plibersek and former NSW education minister Adrian Piccoli,a National,both dubbed the pot of money a"slush fund".

Mr Piccoli,now the director of the Gonski Institute for Education at UNSW,called the $4.6 billion package"pathetic".

"There is nothing fair about it,"he said."There is nothing Christian about it. It’s throwing money at the powerful and well connected."

Blaise Joseph,policy analyst at the right-leaning Centre for Independent Studies,described it as a"special deal"that undermined the Gonski 2.0 funding model.

Mr Tehan conceded the $1.2 billion fund"sits outside the model to deal with those issues that no model is ever going to be able to deal with",and said it was a"wonderful initiative".

There is nothing Christian about it. It's throwing money at the powerful and well connected.

Adrian Piccoli,former NSW education minister

The most expensive element of the policy announced Thursday is a transition fund to help schools move to a new method of assessing their socioeconomic status (SES) score and therefore funding.

Currently,a school's individual SES score is determined by census data from the surrounding area. Under the new model,it will be calculated based on the income tax returns of parents. It is made possible by improvements in the government's data-matching capability.

In total,private schools in both NSW and Victoria will receive an extra $1.46 billion by the time the transition is complete in 2029.

Asked about the absence of any new money for public schools,Mr Morrison said public school funding was chiefly the responsibility of state governments.

"I don't think parents will be cynical,"he said."They will note that we're funding public schools at record levels. And all parents want to have choices about how they educate their children."

The Catholic and independent school sectors cautiously welcomed Thursday's announcement,but left the door open to future arguments over funding.

Ray Collins,acting executive director of the National Catholic Education Commission,said the new SES model was"fairer"and went"a long way to fixing problems caused by the model introduced in 2017",but it would need to be reviewed again.

Michelle Green,chief executive of Independent Schools Victoria,said the deal created"potential"for a workable long-term solution.

Ms Plibersek said Labor would go even further and restore all of the funding it promised under the original Gonski arrangements,which would mean more money for Catholic,independent and especially public schools.

"We don't believe this restores all of the money that has been cut from Catholic and independent schools,"she said.

"This is still billions of dollars less than all schools would have got under the original arrangements struck by Labor."