City have so far proved unwilling to match Brighton's £50 million valuation for the 24-year-old, who handed in a transfer request last week amid concerns his transfer fee is too high.
However, Brighton have stood firm and the lack of progress in talks with City has given Chelsea optimism a deal could be reached.
Chelsea are scouring Europe for defensive reinforcements having missed out on several targets as they seek to replace Antonio Rudiger and Andreas Christensen, who left on free transfers for Real Madrid and Barcelona respectively.
Cucurella is thought to be open to a move to Chelsea with head coach Thomas Tuchel viewing the Spain left-back as a possible left-sided centre-back.
Sources have told ESPN that Chelsea seriously explored signing another right-sided defender in a similar mould, Djed Spence, prior to his move from Middlesbrough to Tottenham -- even suggesting at one stage they would theoretically match the £19m Spurs eventually paid -- before Chelsea decided not to follow through with a formal offer.
Sources claim Leicester have tried to warn off potential suitors by valuing Fofana in the region of £80m.