Los Blancos entered the market for a new stopper when it was confirmed that Thibaut Courtois had suffered a serious knee injury during a training session on Thursday.
Courtois is expected to be absent for the entire 2023-24 campaign, having damaged his ACL, and the likes of David de Gea and Sevilla's Bono have been linked with moves to Bernabeu of late.
However, Kepa has emerged as Real Madrid's leading target, with Carlo Ancelotti's side moving quickly for the Spaniard amid claims that Bayern Munich were also keen on his services.
On Saturday, it was reported that the 28-year-old was preparing to join Los Blancos on loan for the season, although there would not be the option of a permanent deal.
Kepa has been left out of the Chelsea squad for Sunday's clash with Liverpool in a clear indication that he is on the verge of leaving the club.
According to transfer expert Fabrizio Romano, the Spain international has already said his goodbyes to his Chelsea teammates, and Real Madrid will announce a deal 'very soon'.
Andriy Lunin started Saturday's 2-0 win over Athletic Bilbao, and Los Blancos head coach Carlo Ancelotti praised the stopper for his performance during his post-match press conference.
"He's done really well. He was very calm and focused. He moved the ball well from the back and he was good with the high balls. He showed confidence and concentration," said the Italian.
However, Lunin is expected to be replaced in the team by Kepa, with the former Athletic Bilbao said to be moving to Bernabeu as the number one.
The Spaniard started his professional career at Athletic, making 54 appearances for the Lions before moving to Chelsea in 2018, with the Blues paying a world-record £71.6m fee for a goalkeeper.
Kepa has made 163 appearances for the capital outfit, including 39 in all competitions last term, but he has been allowed to leave his parent club this summer.
Providing that there are no late issues, the stopper could make his debut for Ancelotti's team in their next league match against Almeria on August 19.
Chelsea have already signed Robert Sanchez from Brighton & Hove Albion this summer but are now believed to be considering moving for Nice's Kasper Schmeichel to further boost their goalkeeping department.