The 21-year-old goalkeeper signed for the Blues in 2011, but is yet to have made an appearance for the West London club.
Upon his arrival at Stamford Bridge he was immediately loaned to Atletico Madrid, where he has since stayed.
His performances in Spain have been impressive, alerting a host of the division’s top sides.
Real are believed to be particularly keen, as they seek to find a long-term replacement for out of favour captain Iker Casillas.
Despite Courtois currently playing for their bitter cross-town rivals, Los Blancos are ready to lure him to the Bernabeu, which was initially a plan put into action by Jose Mourinho while he was in charge of the club.
A source close to the giants of Spanish football said that interest in the glovesman is serious:
“As soon as Courtois arrived on loan at Atletico from Chelsea, a file was opened on him under instructions from Mourinho.
“The information built up is exhaustive and Courtois is on Madrid’s list of targets, despite the presence of Iker Casillas and Diego Lopez in Carlo Ancelotti’s squad.”
Chelsea are understood to be reluctant to let their promising ‘keeper leave, but his desire to play regularly may be a decisive factor in any potential move.