The Turnbull government may have found an unlikely ally on education funding, after the Greens left the door open to supporting the government’s new 10-year schools funding package.

And despite complaints from a furious Catholic education sector, the fate of 24 ‘overfunded’ wealthy private schools looks all but sealed, with Labor offering its support to cut their funding.

With the Greens’ nine Senate votes, the government would only need the support of one more senator to pass its new funding model – dubbed Gonski 2.0 – which could help avoid a more lengthy negotiation with other crossbenchers or Labor.

The Greens appeared to signal support for the government’s plan on Wednesday, with leader Richard Di Natale saying his party wanted to end “the ongoing war based around how we invest in our education system”.

Greens education spokeswoman Sarah Hanson-Young said ‘overfunded’ schools should lose money to secure a needs-based funding model.

She said some private institutions were building new swimming pools and rowing sheds while public schools were “struggling to even pay their teachers on an ongoing basis”.

“We’ve got to put equity back into the system,” Senator Hanson-Young told Sky News.

Labor, which has otherwise slammed the schools package as a “$22 billion cut” to schools, also signalled it may be able willing to pass elements of the plan.

“If Mr Turnbull wants to cut the funding of 24 elite schools and freeze the funding of 300 elite schools, we are up for that, fair enough,” Opposition Leader Bill Shorten said.

“If we proposed it, the Liberals would have gone off their trolley. If he is proposing, it we’re happy to work with him on that.”

But Mr Shorten also sided with the Catholic sector’s complaints: “It is not me saying that fees will go up … It is the people who run the Catholic education system.”

Former prime minister Tony Abbott flagged possible discontent within the Coalition, saying the proposal would be “pretty vigorously debated in the party room next week”.

“It’s hard to be against extra funding for schools, but I note that at this stage it’s hard to see that any of this extra funding is specifically tied to better academic outcomes and better student performance,” he told 2GB.

Crossbench kingmaker Nick Xenophon and One Nation are waiting to see the legislation, with the government to brief them on the changes next week.

Education Minister Simon Birmingham was forced to defend the cuts to 24 wealthy ‘overfunded’ schools, which drew a furious response from the Catholic sector on Tuesday.

“Unsurprisingly, people who didn’t get everything they asked for might sometimes complain about the consultation,” Senator Birmingham said.

“That doesn’t mean there wasn’t consultation. It just means that we are not going to continue with a system that is based on special deals for one state or another state, one sector or another sector.”

Aside from the 24 schools that will lose funding, a further 353 independent and Catholic institutions will receive less money than they otherwise would have. Those reductions would be used to pay for a $18.6 billion overall funding boost over 10 years.

The National Catholic Education Commission has said schools will have to hike fees or cut costs by reducing staff, while the Catholic Education Office warned some institutions in the ACT may close.

“All yesterday’s announcement has done is provide more uncertainty for the 1731 Catholic schools in Australia,” acting executive director Danielle Cronin said on Wednesday.