Spanish newspaper Marca claim the Belgium international has notified friends and family that a move is on the cards.
The Chelsea keeper has been linked with a move to the Bernabeu since returning from his successful three-season loan spell at Atletico Madrid in 2014.
Real have had their transfer embargo reduced to just one window and are already plotting their summer business.
“Is Thibaut Courtois heading to Real Madrid?”
Chelsea Transfer News
The La Liga giants only have Keylor Navas and Kiko Casilla in Zinedine Zidane’s first-team squad.
Courtois is under contract until 2019 – and the Blues will be reluctant to part with one of their star performers.
Victor Lindelof said goodbye to Benfica fans on Wednesday night.
That’s according to Portuguese paper A Bola who say he was the last man to leave the pitch following their 2-0 win over Rio Ave.
The Sweden star is widely expected to complete a £38m move to Manchester United in the next few days.
It is possible it will be done and dusted ahead of the window opening on January 1, however he can not be registered as a player for Jose Mourinho’s side until then.