Both European heavyweights are looking for emergency goalkeeper backup before the transfer window closes, with Manuel Neuer still fighting after fracturing his leg in a horrific ski accident, and Thibaut Courtois set to miss the season due to an ACL injury.
Kepa was first linked with a surprise move to Bayern, with Neuer reportedly back in training but unable to kick a ball.
However, following Courtois’ injury, Real Madrid entered the sweepstakes, and the Spaniard is eager to join the La Liga giants.
Real have also been in contact with David De Gea, a free agent after leaving Manchester United at the conclusion of last season.
Real look to be debating whether to pay a large amount for Kepa, who has two years left on his deal with Chelsea, or save some money and go with the 32-year-old De Gea instead.