And the Blues have also finally agreed a £38million fee with Real Madrid to sell wantaway star Thibaut Courtois.
Kepa will now undergo a medical at Stamford Bridge today ahead of making the move which will see him become the most expensive keeper in history.
The 23-year-old flew to London yesterday evening and will spend the day completing the formalities of the deal, which Chelsea hope to announce later today.
And once they formally complete, they will then allow Courtois to fly to Madrid to undergo his own medical ahead of his switch to the Bernabeu.
The Blues made their move for Spanish international Kepa over the weekend having finally conceded defeat in their attempts to convince Courtois to stay.
When the Belgian refused to report for pre-season training this week, it forced the club to get a deal done quickly before the transfer window closed so they went straight in with the fee needed to trigger Kepa's release clause.
Ironically, the Spanish number two was on the verge of signing for Real earlier this year for just £17.9m – only for it to fall through at the last minute.
Kepa claimed he rejected the move in favour of staying at Bilbao to continue his development, although there were suggestions ex-Real boss Zinedine Zidane blocked the deal.
The fall-out from that failed transfer saw the keeper sign a new seven-year contract with Bilbao, which then inserted the new £71.6m release clause Chelsea have now had to pay.
Representatives for Kepa arrived at the La Liga headquarters in Madrid early this morning to formally lodge the funds, which will then be forwarded on to Bilbao when he completes the move.
At that point Courtois, who is currently at home in Belgium, will then be given the green light to complete the move to Real he has been agitating for.
The 26-year-old has agreed a six-year deal with the European champions, believed to be worth £200,000-per-week.
Kepa will be joined at his Chelsea medical by Real Madrid midfielder Mateo Kovacic, who is close to completing a season-long loan switch to Stamford Bridge.
A deal for the Croatian midfielder was struck when officials at Chelsea and Real were thrashing out the terms of the Courtois deal.
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