The Belgium international is currently under contract until the summer of 2019 but his future is in doubt.
Courtois, who replaced Petr Cech as the club's number one in 2014, has made 106 league appearances for the Blues so far.
He helped them win the Premier League title last season but is now thought to be open to moving on to join the Spanish giants.
Courtois enjoyed a successful loan spell at Atletico Madrid earlier in his career and reports claim he is stalling on discussing a new contract from the Blues.
Real Madrid are planning to make a bid next summer but Chelsea are ready to let him leave for free in two years time if they can not agree an extension.