The Spanish giants have been linked with a swoop to sign the Belgium international throughout the summer transfer window so far.
Courtois is keen to secure a return to the Spanish capital following his three-season loan stint at Atletico Madrid between 2011 and 2014.
The Chelsea FC number one is eager to secure a return to Madrid as los Blancos look to replace Costa Rica international Keylor Navas.
The Blues goalkeeper has less than 12 months to run on his current deal and Courtois could leave for nothing next summer.
Belgian reporter Terreur shed light on Real Madrid’s bid to sign Courtois ahead of England’s transfer deadline day.
Terreur wrote on Twitter: “Real Madrid have tabled an offer of +/- £35m for Thibaut Courtois. Chelsea working hard on finding a replacement for the AWOL GK – although Real are pushing since the last week of the World Cup. #cfc”
Chelsea FC finished in fifth spot in the Premier League table last term to miss out on a spot in the Champions League.
The Blues won the FA Cup thanks to a 1-0 win over Manchester United to end Antonio Conte’s final season with a trophy.
The west London side appointed Maurizio Sarri as their new manager following Conte’s dismissal.
Chelsea FC have signed Napoli midfielder Jorginho in a £57m deal as well as veteran England goalkeeper Rob Green.
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