At least two Liberal senators and a lower house MP are indicating they won't support the $18.6 billion package, or a compromise with the Greens, on the floor of parliament.
"The government remains committed to its model, to its plans," Senator Birmingham told reporters in Canberra on Tuesday ahead of a joint coalition parties meeting.
"I'm not anticipating changes being discussed or adopted in the party room."
The Catholic schools sector claims the overhaul will cost it more than $4 billion, and it has the support of retiring Liberal senator Chris Back.
Veteran Liberal senator Eric Abetz is refusing to back it because he's "very concerned about the Catholic sector" and a possible government deal with the Greens.
"I hope they haven't done a dirty deal with the Greens because that would be absolutely horrific," he said.
Meanwhile, Liberal MP Kevin Andrews has flagged he might also vote against any amended package agreed to with the Greens.
With parliament due to rise at the end of the week for the winter break, the coalition needs 10, or possibly 11, extra votes in the Senate to get its package through.
Senior officials from the Catholic Education Commission met Senator Birmingham on Monday night.
The discussions apparently didn't go well, with The Australian quoting a source saying the minister was told the government would "wear this like an albatross around its neck until the day of the next election".
The Greens have been offered a compromise that includes reducing the rollout timetable from 10 years to six, more accountability on state government funding and an independent watchdog.
The minor party will meet on Tuesday to discuss the offer, which could add up to an extra $5 billion in funding.
"The worst thing we could do is try and see a bill rushed through the parliament. I think the primary consideration is getting it right," Greens leader Richard Di Natale said.
He insisted the party was "on a unity ticket" on schools' funding, despite reports its NSW branch want senators to insist on the original Gonski deal struck with the Gillard Labor government.
The party's decision could be swayed by reports of a split within the teachers union, which has been staunchly against the changes even though they benefit public schools.
The WA division says the package should be passed if it compels states to meet their funding obligations because it gives a better deal for that state, The West Australian newspaper reports.
Labor won't support the coalition plan, arguing it represents $22 billion less than it had planned for when in government.
Treasurer Scott Morrison said the government's commitment was not fake funding unlike Labor's plan.
"What we've got is real funding," he told reporters.
