The two powers have already drawn up a deal that could be signed in the following days, a Whitehall source has confirmed to the PA news agency.
The UK's associate membership of the €95bn (£81bn) Horizon Europe programme was foreseen in the 2020 Brexit agreement. However, the EU has been delaying the UK's membership due to the dispute over the Northern Ireland Protocol.
Nonetheless, the signing of the Windsor Protocol in February has seemed to solve the disagreements.
The new draft would reportedly allow the UK to join the research programme. However, sources have pointed out that the cost of the membership could be a point of contention, and delay or even block negotiations.
"The EU have moved a little on the money but not nearly as much as the PM wants," a source told PA.
The government did not deny a draft deal had been written, but a spokesman said: "Talks are ongoing and therefore, we have not yet agreed a deal."
In addition to Horizon Europe, the UK is yet to obtain formal access to Copernicus, the Earth observation programme on climate change; Euratom, the nuclear research programme, and services such as Space Surveillance and Tracking.
According to Politico's sources, the UK could choose to rejoin Horizon Europe and Copernicus for the period 2021-27, but not Euratom's nuclear energy R&D scheme, which officials consider "poor value for money."
A deal on the subject would be considered a relief to the British scientific community, which has greatly been affected by the lack of access to Horizon funds. Last year, the Foreign, Commonwealth and Development Office (FCDO) said the EU's refusal to finalise access is "causing serious damage to research and development in both the UK and EU member states".
Greg Clark, the Commons Science, Innovation and Technology Committee chairman, said: "The reports that an agreement may soon be reached on the UK re-joining Horizon Europe are a welcome sign of progress. We look forward to examining any agreement that is reached."
Professor Sarah Main, the executive director of the Campaign for Science and Engineering, said: "This is an encouraging development. If confirmed, this raises the promising prospect of a deal within days, which would be the best of news for UK science.
"I urge the Prime Minister and president of the European Commission to secure this agreement, enabling researchers to get on with the vital role they play in improving people's lives and livelihoods."
UK Prime Minister Rishi Sunak is scheduled to meet European Union chief Ursula von der Leyen in Lithuania next week, providing an opportunity for in-person talks on Horizon.
If an agreement is reached, a formal announcement is expected on Tuesday.
