Belgian ace Hazard, 27, had encouraged the Spanish giants to sign him during the World Cup Finals – but the two clubs remain poles apart with their valuations.
Real are willing to pay £150m for Hazard, who has so far refused to sign a £300,000-a-week extension to his deal which runs until 2020.
But Chelsea want more than the £198m that took Brazilian superstar Neymar from Barcelona to Paris Saint-Germain last summer.
Real’s top brass feel that is just too much for a player who has only two years left to run on his current contract – which is why they have reluctantly decided to give up on their pursuit.
Hazard is set to be offered the Chelsea captaincy as bosses look to persuade him to agree to the biggest contract in the club’s history.
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New Blues boss Maurizio Sarri is desperate to keep hold of his star men this summer.
Hazard, Thibaut Courtois and Willian have all been linked with moves away from Stamford Bridge.
Real Madrid are in the hunt for Belgium No 1 Courtois, while Willian, 29, is a target for Jose Mourinho at Manchester United.
Sarri met with owner Roman Abramovich and director Marina Granovskaia here in Nice this week to discuss transfer targets.
Juventus duo Daniele Rugani and Gonzalo Higuain and free agent Bernard top the Italian’s wish list.