In a submission to the House of Commons Education Committee, three of the university’s pro-vice chancellors have urged the government to recognise the "economic and educational benefit of free movement of talent" and to remove international students from any net migration target.
The submission is in response to a parliamentary inquiry on the impact of Brexit on higher education.
The university asked for greater clarity on the government's plans regarding EU student finance, and for transitional arrangements to be put in place to ensure that students who start before Brexit will continue to have the same financial status as UK students for the duration of their courses.
The government also needs to clarify the status of EU staff working in the UK "as a matter of priority", the statement said. The current uncertainty is affecting the competitiveness of the sector, and the government should consider a transitional arrangement to help universities to attract and retain staff.
Despite the risks, Cambridge recognises that Brexit offers an opportunity to review the visa system, and "ensure that the UK is able to recruit the brightest and best students and academics from around the world", it said.
As well as removing overseas students from immigration figures, the university asked that the government "radically rethink" the tier 2 immigration route for staff, and "look at how we can reduce the administrative and cost burden of visa applications for PhD level staff posts".
Cambridge currently admits around 350 non-UK EU students a year, which is around 10% of its undergraduate intake.
"This is an increase of a quarter on the position a decade ago, and has been driven largely by applications from EU accession states," it said.
The latest data on undergraduate applications for admission in 2017 shows a drop in applications from the EU, despite an increase in applications overall, the university said. EU applications were down by 14.1% from 2,651 to 2,277 while overall applications were up by 3.2% from 16,368 to 16,899.
UK universities can currently offer lower fees to EU and UK students than to non-EU students, but after Brexit it is likely that it will become unlawful to offer a lower fee to EU residents, Cambridge said.
"Assuming that EU students move to the unregulated international rate it is almost certain that application numbers will fall further. We are currently modelling a two-third reduction in admissions from the non-UK EU," it said.
"Any changes to the fee status of EU students are likely to have a significant impact on … our ability to attract top students from Europe. In addition, an increase in international recruitment due to a reclassification of EU students would increase demand for a limited number of overseas undergraduate scholarships," it said.
Employment law expert Euan Smith of Pinsent Masons, the law firm behind Out-Law.com said: "Universities across the UK have been seeking legal support as a result of the levels of concern among staff and students about the implications of Brexit. What’s clear is that the lack of certainty about the future is causing people to question their plans and to reconsider whether committing to the UK in the long-term is a viable option for them. The sooner that firm proposals are made, the better the position for everybody."
“We’ve also seen a notable increase in the number of inspections carried out by UK Visas and Immigration in universities to ensure that they are complying with their immigration obligations. It would not be surprising if that were linked to the government’s previously stated position that they see student visas as being abused as a means for non-EU nationals to come to the UK for longer than they should," Smith said.