The Spanish youngster tops the summer shopping list of Chelsea owner Roman Abramovich and Diario Gol says Real fear Asensio has already been unsettled.
Chelsea have assured Asensio they'll double his current €4.5m salary and are prepared to go as high as €150m to bring him to London.
Real management are desperate not to lose Asensio, but believe Chelsea's move has turned the player's head.
Asensio has made it known to Real chiefs that he will push to leave if a star signing is made to compete for his position. The midfielder is also insisting on a pay rise to stay in Madrid.
His priority is to remain with Real, but Asensio will seek a move away if his two demands are not met.