Los Blancos are lining up the Italian chief, 49, to replace under-fire Julen Lopetegui after his horror start to life at the Bernabeu.
Real Madrid sit seventh in La Liga with just four wins in nine league matches this season.
Galacticos bosses are now mulling over the possibility of a quick fix – with Conte the target.
However, reports in Spain claim Chelsea would be owed a whopping £9.5m in compensation as part of the deal after he was given the boot by the Blues this summer.
Meanwhile, Real Madrid are concerned they may be beaten to Conte's signature by Manchester United, according to AS.
Another former Chelsea manager, Jose Mourinho, is currently running the show at Old Trafford.
However the self-proclaimed Special One has found results tough to come by this season – leading United to a lowly tenth after nine games.
Oddly enough, both Lopetegui and Mourinho have seen their teams win four, draw two and lose three of their opening nine league clashes.
And both could now be under-fire – with Conte seemingly the man to save both clubs and Chelsea licking their lips at the prospect of a bumper pay-day.
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