Leading private schools have challenged plans to widen access to the most selective universities in England, warning they could lead to discrimination against young people “on the basis of the class they were born into”.

The intervention by the Headmasters’ and Headmistresses’ Conference (HMC), which represents many of the country’s most expensive independent schools, reflects members’ concerns that new measures to improve access for the most disadvantaged students could lead to discrimination against students from elite private schools.

The HMC was responding to plans, published on Wednesday by the higher education regulator for England, the Office for Students, that promise to halve the access gap at England’s most selective institutions in the next five years, increasing the number of disadvantaged students by 6,500 each year from 2024-25.

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HMC’s executive director, Mike Buchanan, said universities should expand to accommodate as many “truly suitable students” as necessary, rather than “rob some students of a future to award it to others”. He also called on universities to review the increasing number of international students, rather than “deny places to UK students based on their class”.

Since the lifting of the student admissions cap in 2015, the number of places at many universities has grown exponentially, but undergraduate numbers have remained relatively stable at Oxford and Cambridge, though both claim to have made significant progress in diversifying their student body.

The OfS report, Transforming Opportunity in Higher Education, details ambitious commitments by universities to improve equality of opportunity for students. Currently, young people from advantaged areas of England are more than six times as likely to attend selective universities, including Oxford, Cambridge and other members of the Russell Group, as those from disadvantaged areas.

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Despite the huge expansion in university places, the gap has barely changed. Under new plans drawn up by universities and approved by the OfS, the ratio will be cut to less than 4:1 by 2025, and could be eliminated in 20 years. The OfS is also hoping to reduce the gap between the proportion of white and black students awarded a top degree – a first or a 2:1 – from 22% to 11%.

Buchanan said the HMC was confident their students would continue to secure places at prestigious universities at home and overseas, thanks to excellent results and soft skills. “However, care is needed in starting actively to discriminate against individual young people on the basis of the class they were born into. The country needs all its young people to reach their potential if we are to create a bright new future for Britain post-Brexit.”

Buchanan said contextual admissions – which allow universities to take into account an applicant’s educational and socio-economic background – were reasonable “if used on a sophisticated, individual basis”, but it should not be about school type.

“Independent schools play an important role in getting disadvantaged students into university through offering free and discounted places. Not all state-educated students are disadvantaged and the majority of students from affluent backgrounds are not educated in HMC schools. This is why a sophisticated approach is needed for the country genuinely to level up.”

Kalwant Bhopal, a professor of education and justice at Birmingham University, said: “It is clear that those students who attend independent fee-paying schools are more likely to be white and middle-class and are more likely to go on to hold top high-earning jobs. These schools continue to perpetuate privilege. Contextual admissions are one small step to addressing inequalities of opportunity facing children from many working-class, and black and ethnic minority families.”

Chris Millward, the OfS director of fair access and participation, said if universities did not increase student numbers, then groups that were currently highly represented would end up being less represented as a result of the new targets.

“We expect providers to work towards these targets because they tackle two urgent priorities: the need to open up all of our universities to people from those communities where progress into higher education is lowest, and to ensure that every student has the same chance to succeed once they get there.”

The universities minister, Chris Skidmore, said it was damning that such large gaps still remained between disadvantaged students and their peers. “I am pleased to see universities being ambitious in their plans to reach out to those from disadvantaged backgrounds and to support them through their studies. But for universities which do not meet their registration conditions, I fully support the OfS to take appropriate action.”