If he does leave, though, there seems to be a lack of obvious options to replace him.
Carlo Ancelotti is the bookmaker's choice by a mile, and the efforts of his PSG side against Barcelona in the Champions League recently will have done him no harm, despite elimination, in the eyes of Florentino Perez.
Then there's the German pair, Joachim Low and Jurgen Klopp, who would both likely be welcomed by not just Sami Khedira and Mesut Ozil but also by the Los Blancos supporters.
Both appear unattainable now, though. The start of next season marks a World Cup year for Low, who manages Germany, and Klopp has expressed his intent to do battle with Josep Guardiola's Bayern Munich (via Goal.com).
One of the most intriguing outsiders is Swansea manager Michael Laudrup, the Dane who recently lifted the League Cup in England with the Welsh side. The fact that he dropped the Barca dream team for Madrid in the '90s would no doubt count in his favor too.
Not only was the former Danish international a successful player in Spain, but he's also had two spells in management there—one in the city of Madrid.
In the Spanish capital, he managed Getafe—with an expansive attacking style similar to that now employed at Swansea—to memorable wins over Barcelona and Real Madrid as well as a UEFA Cup quarterfinal and the final of the Copa del Rey.
However, his second stint in Spain, with Mallorca, toiled to the backdrop of boardroom squabbles and broken promises.
Other managerial spells have included Brondby, where he began his coaching career with a wining rate of close to 60 percent, and a brief stint at Spartak Moscow, which ended unsuccessfully after seven months.
In The Mirror in February, he seemed more than open to thoughts of sitting in the dugout at the Santiago Bernabeu:
It is a pleasure that the fans think that I could be the next Real Madrid manager, we will see what happens in June or July...It's certain that I will return to Spain, but I don't know when...Spain is like my second home, but I am enjoying my experience in the Premier League.
At the age of 48, he can still be considered a young manager, but he's old enough to have gained enough experience to be considered a candidate for the Madrid role.
His playing career dictates respect from the players whom he manages, and he's still not bad with the ball at his feet:
How seriously Perez would take him his unknown.
His standing in the game is worthy of a Madrid manager, and his recent success with Swansea, as well as his impressive records with Brondby and Getafe, tick most of the boxes.
It would, of course, carry an element of a gamble, but what managerial appointment doesn't?
As an outside runner, Laudrup's not a bad bet.