Anti-Brexit demonstrators wave EU and Union flags outside the Houses of Parliament in London Thomson Reuters By Samantha Koester

BRUSSELS (Reuters) - Britain's exit from the European Union may hurt one of the most popular programs launched by the EU -- the Erasmus student exchange -- cutting the number of available destinations and participating students, officials said on Tuesday.

The country's eligibility for the program could run out in December 2020 when a transition period ends after Britain officially leaves the EU in March 2019, unless it and the bloc negotiate a continued participation.

Without a deal, British universities would no longer be available for exchange to students from the other 27 EU countries, nor would be universities in the EU for young Britons.

Erasmus, under which some 725,000 EU students annually go to study in another EU country for up to a year, is funded by the European Union from a budget of 14.7 billion euros allotted for the period between 2014 and 2020.

Britain is the third most popular destination for EU students in the scheme after Spain and Germany, Naquita Lewis, in charge of the Erasmus scheme in Britain, told a seminar.

In 2015 some 30,000 young people went to study in Britain as part of the program, she said, while some 40,000 Britons traveled to learn in other EU countries.

"It is going to be hard to keep the framework of Erasmus if we become not part of the EU," Erasmus Student Network President Joao Pinto told the seminar, organized by the Centre for European Policy Studies and the British Council.

British Prime Minister Theresa May said at an EU summit in December that Britain would remain part of the scheme "at least" until the end of 2020, but it is unclear what happens later.

All EU countries have full access to the Erasmus scheme but so do several non-EU countries like the former Yugoslav Republic of Macedonia, Iceland, Liechtenstein, Norway and Turkey.

The lack of eligibility of British students for Erasmus after 2020 could be felt the most by minority students or those who cannot afford to study abroad on their own, officials said.

"Mobility is not for all, and we actually use Erasmus to level the playing field," said Isabell Majewsky Anderson, head of the Go Abroad office at the University of Edinburgh.

(Reporting by Samantha Koester; Editing by Jan Strupczewski and Hugh Lawson)