The government can pass its changes into law without the support of the Greens after it secured the necessary 10 votes from the Senate crossbench on Tuesday.

The victory will be a massive blow to Labor, the Catholic education sector and the Australian Education Union, which all campaigned ferociously against the government's new funding model.

To seal the deal, the government agreed to major concessions including speeding up the delivery of the package from 10 years to just six.

This will add $4.9 billion on top of the $18.6 billion in extra spending the government had already announced.

The government will also introduce new rules forcing state governments to increase their spending on public schools and will create an independent schools resourcing body as recommended by the Gonski report.

The four One Nation senators and three Nick Xenophon Team senators have announced they will support the government, as will Derryn Hinch, Jacqui Lambie and Lucy Gichuhi.

Despite achieving all their major negotiating demands, the Greens remained paralysed by internal division and had yet to announce a final position.

Senator Xenophon said the crossbench had delivered the "real Gonski" rather than the "knock off" version implemented by Labor in 2013.

"This is the real Gonski because we have David Gonski, we have his panel members, saying unambiguously it would be a disaster for Australian education if this package wasn't passed," Senator Xenophon said.

He said the government's agreement to fast-track its spending was a "huge breakthrough".

Earlier on Tuesday, Labor education spokeswoman Tanya Plibersek urged the Senate not to pass a funding model that "parents hate, that teachers hate, that will be terrible for Australia's children".

"This is not needs-based, it's not set the blind, it is not fair," she said.

Opposition Leader Bill Shorten said: "Labor cannot in good conscience support a $22 billion cut to school funding."