RMC Sport report that the 57-year-old is among the candidates being considered by Florentino Pérez, with his contract as France manager expiring after this summer's World Cup in North America.
Deschamps has already made clear he is not retiring from coaching — only stepping away from international football after 14 years in charge of Les Bleus. He guided France to World Cup glory in Russia in 2018, reached the final again in Qatar in 2022, and has long been considered one of the best man-managers in the game.
Real Madrid currently have Álvaro Arbeloa in charge, the former full-back having replaced Xabi Alonso when the Spaniard was dismissed in January. The club has given no indication of whether Arbeloa will continue beyond this season, but with Real facing potential elimination from the Champions League and trailing in the La Liga title race, the pressure on the current set-up is significant.
Three factors make Deschamps an attractive proposition for Pérez.
His track record of winning major trophies speaks for itself — Champions League final with Monaco, Ligue 1 with Marseille, and two World Cup finals with France.
His long-established relationships with key Real Madrid players also count heavily in his favour. Kylian Mbappé, Aurélien Tchouaméni and Eduardo Camavinga are all regulars in the France squad under his management.
His reputation for handling high-pressure dressing rooms and making difficult decisions around big-name players rounds out the appeal.
Deschamps is one of several names under consideration. Jürgen Klopp has ruled himself out. Mauricio Pochettino, currently coaching the United States, and Massimiliano Allegri are also reported to be in the frame.
The French manager has been linked with Saudi Arabia and various European clubs, but has not publicly addressed his next move. Those close to him say all options remain open.
The role reversal element adds a further layer of intrigue. Zinédine Zidane — three-time Champions League winner as Real Madrid manager — is widely expected to succeed Deschamps with the French national team once the World Cup concludes.
One legend heading to Madrid, another to Les Bleus — if the reports prove correct.