Centre Alliance has offered the government a pathway to pass its higher education funding changes, indicating it could shift its position in return for more support for South Australian universities.

The comments from the party’s education spokeswoman, Rebekha Sharkie, come after universities softened their criticism of the Coalition package, prompting outrage from the National Tertiary Education Union which labelled the move “extremely disappointing”.

Although universities were strident in criticism before the job-ready graduate package was referred to a Senate inquiry, Universities Australia has now called for “funding certainty” and submitted that it will not oppose the bill if further minor amendments are made.

The legislation proposes to reduce the overall government contribution to degrees from 58% to 52% and to increase fees by up to 113% for some courses, including humanities, to pay for fee cuts in sciences and 39,000 extra university places by 2023.

The government needs at least one more Senate crossbench vote out of the trio of Centre Alliance’s Stirling Griff and the independents Rex Patrick and Jacqui Lambie – who are waiting for the outcome of the snap inquiry before the bill is reconsidered in October sittings.

In June Sharkie expressed concerns about fee increases for humanities students. On Monday Sharkie told Guardian Australia that although it “couldn’t support the bill as it currently is”, Centre Alliance was seeking changes on behalf South Australia’s three universities.

Because all three of the state’s universities are classified as low-growth metropolitan universities, the package offers them funding for a 1% growth in bachelor places, compared with 2.5% for high-growth metro universities and 3.5% for regional universities.

“There are some parts of the bill that have merit but what’s proposed for South Australia is concerning,” Sharkie said. “The way it’s designed seems to assume that every state has both regional and metro unis.

“We’re a small state – our regional kids come into the city to the three unis, which are all metro.”

Sharkie said South Australian universities should get regional status, like the University of Tasmania, because the state has higher unemployment than many other states and is classed as regional for immigration purposes.

“We’re still talking with the government and we’re keen to hear from unis,” she said.

“Universities are mindful they made a strong case against 2017 funding cuts, which the Senate didn’t support, then the government made non-legislative cuts and the [commonwealth grants funding] freeze really hurt them.

“They think it might be a case of better the devil you know.”

The education minister, Dan Tehan, told ABC TV he had had “very good discussions” with the crossbench and confirmed Centre Alliance wants regional loadings applied to South Australia.

In its submission to the inquiry, Universities Australia claimed to neither support nor oppose the bill but asked for “funding certainty” and indicated it would settle for indexation of grants to universities and a review of funding arrangements after three years.

The Group of Eight leading universities set a higher bar but said it would support the bill if the government agreed to curb the biggest proposed fee hikes. It called for “more equitable student contribution amounts”, modifying the structure so no fees are rose above the current maximum rate of $11,355.

That would curb fees set to rise to $14,500 for courses in law, economics, commerce, social studies, political science and communications.

The NTEU national president, Alison Barnes, said the job-ready graduate package “amounts to a funding cut across the board of at least 15% per student place” and cut commonwealth grants by about $1bn a year at current enrolment levels.

“It is extremely disappointing that our vice-chancellors are prepared to accept this, especially in light of no real government assistance with the funding and jobs crisis caused by Covid-19,” she said.

Barnes called for the Senate to send the government “back to the drawing board, to design a system that properly funds our universities so they can truly play a vital role in rebuilding Australia’s economy and help create a better future for all Australians”.

In further submissions, many universities including Edith Cowan, Southern Cross and Federation University rejected plans to remove federal funding from students who fail more than half their courses in their first year.

Nevertheless, Southern Cross said it supported the “ethos and aspiration of the bill”. Federation asked the Senate to pass it to “allow for a timely implementation of the package”.

La Trobe University recommended that the bill pass but asked for a “a reworking of the student [fees] with a view to narrowing the gap between the highest and lowest student contributions”.

Edith Cowan called for more funding to meet additional demand for university places and for the Department of Education to permanently reallocate places from universities that enrol below their caps to those oversubscribed.