For-profit providers have pressed the government to give them greater access to publicly funded student loans and open up teaching grant in high-cost subjects.
Universities have hit back by arguing that only charities should have access to state loans.
Responding last year to the government consultations on the higher education White Paper, private providers also called for university representatives on the Higher Education Funding Council for England board to be replaced with members independent of the sector.
Meanwhile, one major US for-profit, Laureate Education, told the government it would welcome the opportunity to establish or buy a UK university.
Responses to the White Paper technical consultation, obtained by Times Higher Education under the Freedom of Information Act, expose the battle lines drawn between universities and private providers.
The White Paper announced plans to create a "level playing field" between both camps.
Under the proposals, private providers would receive equal access to public funding from the Student Loans Company (instead of the present lower cap of £6,000) but would be obliged to submit to the same regulation as universities.
Although the plans were delayed after the government shelved its higher education bill earlier this year, ministers are expected within weeks to launch a separate consultation to bring private providers seeking SLC funding within the student number cap - and may propose allowing them access to the full £9,000.
Choice means money
In its response to the technical consultation, Laureate tells the government that a "genuine choice" for students can be achieved only "if all providers have an equal right to access to public funds - both funding for students and for teaching grant".
The White Paper proposes that access to teaching grant for high-cost subjects be limited to non-profits. But Laureate highlights its provision of medicine and dentistry education around the world, and asks why it should be "barred" and "thereby be unable to bring that global expertise and capacity" into the UK higher education system.
Laureate also says: "We would support any changes that make the establishment, acquisition, ownership and operation of a licensed and recognised [higher education institution] in the UK more achievable."
Meanwhile, the response from Pearson, a FTSE 100 company that has made no secret of its desire to gain degree-awarding powers, says that "our contribution will be strongest if students studying with us are able to access government student loans".
However, the company raises concerns about being subject to the student numbers cap.
Instead, it proposes a "mechanism" whereby it would cover the cost to the taxpayer of subsidising student loans, estimated by the government at 30 per cent of loan value.
"This plan would enable an increase in the number of (loan-supported) higher education places available for UK students at a far reduced risk and overall cost to the public purse," it says.
Discrimination and exhibitions
The New College of the Humanities, which charges fees of £18,000 and is the subsidiary of a for-profit parent, says that a £6,000 or £9,000 fee cap "will severely limit fair access to premium institutions".
It proposes that in some cases it could offer an "exhibition" - a type of scholarship - to cut fees by £10,800, leaving a net fee of £7,200 that could be covered by an SLC loan.
A solution is needed to "prevent discrimination against those students who would like to benefit from the outstanding quality of teaching on offer from new premium providers", NCH says.
However, universities note that unlike for-profit providers, they are required to meet a public benefit test as charities.
It is "not appropriate for public funds to be invested where there is no guarantee of public benefit, and where they would constitute a subsidy to the profits of owners and shareholders", the response from Durham University says.
"One option may be to make charitable status a condition of the receipt of public funding."
The University of Birmingham says that the problem with the government's proposals is that alternative providers would have access to public funds "when they have no corresponding obligations in relation to public benefit and the long-term interests of the sector".
It echoes Durham's proposal that to access public money, institutions should have charitable status.
Meanwhile, the privately funded College of Law says that the Hefce board "should be restructured to provide a majority of members who are independent of the higher education sector, and able to represent a wider public interest".
Laureate also says of the Hefce and Quality Assurance Agency boards that "all the key regulatory bodies need to reflect [the] broader supplier base".