“We’re writing to alert you to the growing alarm among Independent schools caused by the disturbing signals emerging from public and political discussion about school funding,” they wrote.

Parents would react with “dismay and anger” if the government gave in to a political campaign by Catholic schools to extract a special deal, wrote Independent Schools of NSW chief Geoff Newcombe, Independent Schools Victoria chief Michelle Green and Independent Schools of South Australia Carolyn Grantskalns.

“We support more funding for all schools, regardless of sector, as long as there is a level playing field,” they said.

“This recent campaign, however, has used the rhetorical stereotypes of class warfare, impugned the integrity of staff in independent schools, and published ‘hit lists’ of selected independent schools.

“It would be a backward step if, as a result of this political pressure, we return to the funding wars, in which the stereotypes of ‘class warfare’ and ‘hit lists” re-emerge in practical form.”

The move counters a growing assertiveness from Catholic school authorities in the wake of the byelections last Saturday, where principals at three Catholic schools in the key electorate of Longman emailed parents on the eve of the byelection to influence their votes.

The email to Catholic school parents told them that Labor was offering $250 million more to the Catholic sector across the country over two years – signalling the likelihood of a similar campaign at the general election.

Catholic school authorities used a meeting with Education Minister Simon Birmingham on Tuesday to press for a resolution within weeks with a revised formula to increase their funding.

“It was a productive discussion, but it’s now crunch time for some key decisions to put these issues to bed,” said Dallas McInerney, the chief executive of Catholic Schools NSW.

“We’ll meet again within a fortnight to address the outstanding matters that need resolution.

“We’re trying to ensure the government has a fairer funding model in place for all schools.”

The growing dispute centres on a review of the school resourcing funding model by company director Michael Chaney and others, setting the framework for an attempted compromise with Catholic and independent schools.

Fairfax Media understands the independent school sector fears it could lose $1 billion over a decade under some proposals to help the Catholic sector, tilting the playing field in the competition over fees and services.

There are 1730 Catholic schools educating round 760,000 students across the country.

There are 1061 independent schools educating 604,000 students.

There are 6639 public schools with 2.52 million students across the country.

The letter from the independent schools associations was copied to school principals, clearing the ground for a campaign to warn parents about any loss in funding.

Some principals have already used newsletters and emails to alert parents to the funding problem, ready to match a political campaign from the Catholic sector in the wake of the Saturday byelections.

Government sources told Fairfax Media a resolution could take two months rather than be done in a matter of weeks, as speculated in the media on Monday.

In a serious political danger for Mr Turnbull and the Coalition, any concessions to Catholic or independent schools raise the risk of a more powerful campaign by Labor and the Australian Education Union to warn about funding cuts to government schools.

AEC president Correna Haythorpe said Mr Turnbull should restore $1.9 billion in funding for public education rather than strike special deals.

“Public schools were victims of savage funding cuts under Gonski 2.0, and they must have their funding restored before Mr Turnbull considers any further special funding deals for private schools,” Ms Haythorpe said.