The left-back is tipped for a big future by youth coaches at the Spanish champions and he has been with the club since he was 14.
However, the 18-year-old's current contract expires this summer and he is still yet to agree an extension.
That, according to Sport, has sparked interest from Chelsea and Tottenham and they are now working on deals for Cucurella.
The Premier League pair have scouted the defender during his appearances for Barcelona's youth teams and they have been suitably impressed.
Barca would ideally like Cucurella to continue his development with them, but the youngster may look to following in the footsteps of Cesc Fabregas and Hector Bellerin by leaving Spain for the Premier League as a teenager.