The former Chelsea boss was a near certainty to replace Julen Lopetegui after Madrid renewed contact in the wake of their clásico humiliation at the Nou Camp on Sunday night.
But reports began to emerge on Monday afternoon suggesting that the deal was becoming more complicated, and that Santiago Solari would be taking interim charge at the very least.
Solari, who is currently in charge of Madrid's B team, is seen as a safe pair of hands who could carry the team until either the end of the season or until a further change is deemed necessary.
In Italy, it is claimed that Conte decided against taking the role with Madrid over doubts that he had support of key dressing room figures.
"He has decided, after much thinking time, and the agreement has broken," report Mediaset in Conte's homeland.
Madrid-based newspapers AS and Marca are reporting that Solari is set to take charge of Wednesday's Copa del Rey tie in Melilla and possibly the weekend's fixture with Valladolid.
The Bernabeu club are yet to officially announce the departure of Lopetegui but his impending sack will be confirmed imminently. The former Spain boss is currently with family awaiting his fate.
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