With one Coalition senator now threatening to cross the floor and vote against his colleagues, a deal with the Greens and crossbenchers is the Government's best chance.
Liberal senator Chris Back is retiring at the end of the week.
But in his final days in parliament, he is threatening to cross the floor and vote against the Government on one of its signature policies.
He has told ABC Radio he is concerned Catholic schools will be hurt by the Turnbull Government's education changes.
"My legacy, very strongly, is to support Catholic schools. And, unfortunately, until I'm convinced that the proposals in place will not disadvantage Catholic schools -- and independent schools, for that matter -- then I've indicated to the Minister that, obviously, 'Please don't make me vote against the Government in my last week in the Senate.'"
Senator Back spent nine years on the Catholic Education Commission of Western Australia.
He is pressuring education minister Simon Birmingham to rewrite parts of the Government bill.
Leaked Government modelling suggests Catholic schools will receive billions of dollars less under the deal than under current arrangements while public schools will benefit.
But Senator Birmingham says he remains committed to getting the bill through before the break.
"We really want to see this dealt with this week, because it's important for Australian schools, who are budgeting for next year over the coming months, to have the certainty."
With Labor strongly opposed to the changes, negotiations are underway between the Government and the Greens.
The nine Greens senators have not ruled out backing the plan, which would have the Commonwealth hand schools an extra $18.6 billion over the coming decade.
The Greens are demanding a faster funding rollout and a new independent monitor to regulate the spending.
Greens education spokeswoman Sarah Hanson-Young says there are signs the Coalition is willing to negotiate.
"Unless the Government is willing to come to the table* with those things -- and we've been given some encouraging signs of that -- if we can't make those changes then public schools will continue to miss out. I don't think anyone can argue that 10 years is good enough for our underfunded schools."
Labor, the Catholic sector and the teachers union are urging the Greens to vote against what is known as Gonski 2.0.
But in Question Time, Prime Minister Malcolm Turnbull has continued the sales pitch for the needs-based funding model.
"This is about our children. It's about their future. It's ensuring that they get the needs-based funding they deserve, whatever school they're in, whatever state they're in, whatever system they're in."
Without the Greens, the Government could find the numbers it needs elsewhere on the Senate crossbench.
One Nation's four senators are likely to vote in favour, as is independent Derryn Hinch.
Nick Xenophon says his bloc of three votes is also nearly convinced.
But he is backing the Greens' call for an independent monitor and also wants a speedy result.
"We're not quite there yet, but I think that this has many elements of a good package. It needs improvement, but I don't want the perfect to be the enemy of the good. I think we can actually do some good things this week. So the short answer is, 'Who knows?'"
The bill has already cleared the House of Representatives and is due to be debated in the Senate on Wednesday.
