Sources told ESPN FC that Kepa boarded a flight from Bilbao Airport on Tuesday, with Spanish newspaper AS reporting that he is set to meet with his representatives in Madrid and decide whether to make the move to Stamford Bridge.
On moving to Stamford Bridge, Kepa said: 'It's a very important decision for me, for my career, and also for my personal life.
Courtois has been on strike this week in an effort to force Chelsea to sell him, and the team relented after lining up Kepa Arrizabalaga as his replacement. Mario Stanić was the first, followed by goalkeeper Matej Delač, striker Stipe Perica, and offensive midfielder Marijo Pašalić, though they have all been sent out on loan from The Blues.
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"The Spanish worldwide goalkeeper arrives ahead of the new Premier League season having signed a seven-year contract at Stamford Bridge", Chelsea said in a statement.
"Athletic thanks the player for his contribution during the time he has remained in our club".
The West London side were able to lure the goalkeeper with an improved after-tax salary and a seven-year contract, with Maurizio Sarri ready to hand him a key role in his new-look side.
A move to Real Madrid will mean a return to Spanish football for Courtois, who played for Atletico Madrid before becoming Chelsea's number 1.
Real, meanwhile, confirmed Courtois will undergo a medical on Thursday morning before signing a six-year deal.