Ministers are coming under pressure to lift the limit on the number of places to study medicine in England, after changes to how A-levels are awarded.
The number of students studying to be doctors is regulated because of the cost and for NHS workforce planning.
But universities fear that without the cap on places relaxed and financial support, they can not accommodate all the students with the grades to get in.
The education secretary has apologised for the distress caused by the U-turn.
This is the latest issue thrown up by the government’s decision on Monday to change how exam grades are awarded, following heavy criticism from students, teachers and some Tory MPs.
The move to give A-level and GCSE students grades estimated by their teachers, rather than by an algorithm, means thousands of A-level students may now have the grades to trade up to their first-choice university offers.
In a letter to Education Secretary Gavin Williamson, seen by the BBC, Universities UK sought “urgent assurances” that he was speaking to the Department for Health about increasing the medical student cap.
“The role of universities in training the medical workforce is essential for all regions and nations of the UK, as clearly shown by our members’ response to the Covid-19 pandemic,” it said.
The letter also called more widely for “significant financial support” from the government as students are expected to change courses after being awarded higher grades.
The body, which represents 137 institutions across the UK, said that while the change to the grading method was the right decision, it would lead to grade inflation meaning universities with lower entry requirements would face a drop in course take-up and as a result require financial help.
The letter also asked for clarity on how increased student numbers could be managed alongside social distancing measures and guidance on how to handle a higher number of candidates with the required grades than available places.
Prof Jenny Higham, principal of medical school St George’s, University of London, told the BBC’s Newsnight: “Medicine is both a very practical discipline and also requires a great deal of clinical and practical experience and hence clinical placement capacity also needs to be increased.”
The pandemic has meant the current students have been unable to carry out their clinical studies meaning there is a backlog in places, she said.
Prof Higham added it was a high cost subject with courses funded by supplementary payments from the government as well as tuition fees, and the need to pay for clinical placements.
The BBC has asked the Department of Health for a response.
Conservative MP Sir John Redwood told Newsnight any changes also needed to be fair to the class of 2021 as well as “make up to the class of 2020”, with next year’s cohort needing to be assured of places if they got the necessary grades.
On Tuesday, universities minister Michelle Donelan said she wanted to ensure any students who had accepted a “different course” than planned, as a result of being downgraded last week, should be able to “change their mind and to reverse that decision”.
She said No 10 was working with universities to help “boost the capacity available” in order to “minimise the amount of students that will be looking to defer.”
Ministers in England, Northern Ireland and Wales all decided on Monday – four days after A-level results were issued – to revert to teacher assessed grades rather than the algorithm. Scotland reverted to teacher assessed grades on 4 August after facing a similar backlash.
The move prompted a scramble for university places as students tried to reclaim places at universities which they had last week been rejected from.
However, the top universities warned that students who now have higher grades could still be asked to defer if there is no space left on their chosen course.
The chaos and uncertainty has led to calls from school and college leaders for an urgent review.
“This degree of transparency is necessary at a time when public confidence has been badly shaken,” said Geoff Barton, general secretary of the Association of School and College Leaders union.
Mr Barton also called on No 10 and Ofqual to put in place a “robust contingency plan” for students sitting GCSEs and A-levels next summer in the event of further coronavirus-related disruption.
Meanwhile, pupils will get their GCSE results on Thursday as planned, the Joint Council for Qualifications (JCQ) said.