Courtois has been a Blues player since 2011, when he signed from Belgian club Genk and was loaned to Atletico Madrid.
Chelsea winger Eden Hazard has also been linked with Real, but did attend training on Monday.
Courtois was named best goalkeeper at this summer’s World Cup in Russia, where Belgium beat England in the third-place play-off.
He has played 154 times for Chelsea, keeping 58 clean sheets and conceding 152 goals.
Chelsea were then beaten 2-0 by Manchester City in Sunday’s Community Shield with Argentine Willy Caballero in goal.