Cucurella moved from Getafe to Brighton in the summer of 2021 after the Seagulls splashed £15million on his signature.
The defender enjoyed an eye-catching start to his Premier League career and was even voted as the club's player of the season.
City boss Pep Guardiola was also impressed with the Spaniard, which soon prompted a £30million bid from the Citizens.
Brighton have since turned down the offer, which forced Cucurella to hand in a transfer request as he looked to push through a move to Manchester.
Nevertheless, the Sky Blues, who are looking to fill the void left by the departure of Oleksandr Zinchenko, are reluctant to meet the £50million asking price.
Chelsea, who could have their own gaps to fill should Marcos Alonso and Cesar Azpilicueta complete their moves to Barcelona, have spotted an opportunity and have jumped in the market for the 24-year-old.
According to Sky Sports, Thomas Tuchel's side are willing to meet the asking price set by Brighton and are understood to be in advanced talks.
The Blues have already signed Raheem Sterling for £47million from Man City and Kalidou Koulibaly from Napoli for a further £33million.