Athletic made it clear that they would not accept any bids for Kepa, meaning that the Blues must meet a price tag in his contract in order to proceed.
It is believed that Chelsea have now agreed to pay the fees required to secure the goalkeeper.
The move would break the club's transfer record, set last season when Chelsea signed Alvaro Morata in an initial £58m ($75m) deal, potentially rising to £70m ($90m), from Real Madrid.
It would also make Kepa the world's most expensive goalkeeper, less than a month after Alisson secured that title with a £65m (€73m/$85m) move from Roma to Liverpool.
Chelsea have formalised their interest as they negotiate the exit of Courtois from the club. Goal understands that Kepa's representatives are now in talks with the Blues regarding personal terms.
Real Madrid are keen to secure Courtois in a €35m deal (£31m/$40m), meaning Chelsea will have to find the extra funds needed to sign Kepa from Athletic.
Los Blancos have not yet made a breakthrough in their pursuit of the Belgium international, with Chelsea waiting to have their replacement lined up before sanctioning the deal.
Indeed, the Blues are faced with spending more than they had previously planned to, after missing out on Alisson, who was their first-choice target.