London could make the announcement -- part of Labour Prime Minister Keir Starmer's much-touted "reset" with the bloc -- as early as Wednesday, newspapers the Times and the Guardian reported.
The deal would pave the way for British students to participate in the scheme from January 2027, according to the two papers and the BBC.
Britain left the Erasmus scheme, in which it had participated since 1987, when it left the EU at the beginning of 2021.
The move caused consternation among students and higher education bodies.
Work towards rejoining was announced as part of Starmer's post-Brexit reset deal agreed with Brussels at a summit in May.
The Guardian reported that British students would be able to participate in vocational training placements as well as university-based study exchanges under the Erasmus scheme.
The Times said that the EU had offered the UK government a 30 percent reduction in fees for the first year of membership.
A government spokesperson reiterated the administration's stance of "not commenting on ongoing talks".
But the apparent breakthrough was quickly welcomed by UK universities and Britain's third-largest political party, the pro-EU Liberal Democrats, which called it "a crucial first step" towards a closer relationship with the EU.
The bloc is keen on a wider youth mobility scheme that would allow British and European 18- to 30-year-olds to study and work in the UK and vice versa.
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