"This is about more than just travel: it's about future skills, academic success, and giving the next generation access to the best possible opportunities," the EU relations minister said.
The Erasmus scheme, named after the Dutch Renaissance theologian, was scrapped in the UK in December 2020, when the government announced its post-Brexit trade deal with the EU.
It enables students to study abroad at partner universities and higher education organisations by offering grants to help with living costs.
Participating students usually pay fees to their home institutions, with additional costs covered by the European Union, funded by taxpayers' money.
Britain could have remained a member of Erasmus after Brexit, but then prime minister Boris Johnson said the programme did not offer value for money.
The UK argued that before Brexit more than twice as many EU students came to the UK as British students travelled to Europe at a net cost to the UK taxpayer.
In 2020, the last year in which the UK participated in Erasmus, the scheme received €144m (£126m) of EU funding for 55,700 people to take part in Erasmus projects overall.
The UK sent out 9,900 students and trainees to other countries as part of the scheme that year, while 16,100 came the other way.
In comparison, the Turing scheme - named after British mathematician and codebreaker Alan Turing - received £105m of funding in the last academic year.
This paid for 43,200 placements, with 24,000 of those being in higher education, 12,100 in further education and 7,000 in schools.
Ministers who introduced the Turing scheme in 2021 said it was designed to benefit more people from disadvantaged backgrounds and provide greater support for travel costs than the Erasmus scheme did.
Supporters claimed Erasmus boosted the UK economy even after taking into account membership costs, and helped support universities financially.
Prime Minister Sir Keir Starmer reopened talks in May, claiming that a youth mobility scheme could also be part of a new deal with the EU.
