The Scottish and Welsh governments have expressed fears over the future of the Erasmus student exchange programme in the event of a no-deal Brexit.
Both administrations have jointly written to the UK education secretary to express their concerns.
The EU programme helps students study in other countries.
The UK government has guaranteed payments for successful applicants and said it is "exploring participation" in a successor scheme.
Erasmus is an EU-funded programme which enables students to either study part of their degree or undertake a work placement abroad.
About 53% of UK university students who learn abroad do so through the initiative.
Some countries which are not in the EU - including Iceland, Norway and Serbia - also take part.
The letter is signed by the Scottish higher education minister, Richard Lochhead, and the Welsh education minister, Kirsty Williams.
Both devolved administrations are opposed to a hard Brexit but if the UK leaves the EU without a deal, they would like participation in Erasmus to continue.
The letter argues that leaving the EU without a deal or an arrangement covering the scheme will result in universities, colleges, and schools being ineligible to submit applications to participate in the final year of the current Erasmus+ programme in 2020.
It says: "This will be a significant loss to both our education sectors.
"Between 2014 and 2018, Erasmus+ has enabled an estimate of over 10,000 students and staff in Wales to undertake mobility visits to benefit their learning and career development.
"In Scotland, proportionally more students take part in Erasmus+ than from any other country in the UK."
It calls for an urgent meeting of education ministers to discuss the steps being taken towards ensuring that a hard Brexit does not lead to a loss of provision and opportunities for universities, colleges and schools.
The letter continues: "As we understand the position, the UK government has not, that we have seen, made or planned alternative domestic UK arrangements to ensure that the education sector can continue to access international mobility opportunities in place of those which will no longer be available due to the UK's exclusion from Erasmus."
"Any such plans would, of course, require discussion and agreement with the devolved administrations."
In 2016-17, more than 16,500 UK students participated in overseas programmes through Erasmus while almost 32,000 EU nationals came to the UK through the scheme.
The UK government insists Britain will leave the EU on 31 October but hopes it will be able to do with a deal. Opponents fear a so-called hard Brexit appears an increasingly likely possibility.
Mr Lochhead said: "It is the Scottish government's preference to remain in the EU, but in the event of a damaging 'no-deal' Brexit, students could now see the door to this fantastic cultural and educational exchange slammed shut.
"It is unacceptable that with less than 12 weeks left until the UK government plans to take the United Kingdom out of the EU without an agreement in place, there is still no plan for alternative arrangements."
A Department for Education spokesperson said: "This government is working to negotiate a new deal, but in the event the UK leaves the EU without an agreement in place, we have already guaranteed cover for the payments for successful UK applicants for Erasmus+ and ESC bids.
"Successful bids are those that are approved directly by the European Commission or by the UK National Agency and ratified by the European Commission.
"The UK government has also repeatedly made clear that it values international exchange and collaboration in education, which is why we are exploring participation in the successor scheme and preparing for a range of potential outcomes."