Belgium international Hazard continues to generate intense speculation regarding his future, with a big-money transfer to Santiago Bernabeu having been mooted for some time.
Desailly can understand the appeal to both parties, but feels a key figure for the Blues should embrace his role at Stamford Bridge and avoid becoming another high-profile squad member in Spain.
The World Cup winner told talkSPORT: “It depends on his ambition.
“I’m not sure Eden Hazard has that intention of being that top player, being first choice at Real Madrid.
“Remember, Hazard has been in the Chelsea system for five years, and he’s been doing well, but he’s not been doing what we expected of him – scoring goals and being the equivalent of Ronaldo or Messi, he’s on and off.
“I’m not sure he’s fully ready to change and take the risk, because he will want to play regular football.
“The same thing goes to [Atletico Madrid forward and Manchester United target Antoine] Griezmann - is he really ready, does he have the desire to be in a competition for places at one of the top clubs and not playing football?”
While backing Hazard to stay put, Desailly acknowledges that 18-goal top scorer Costa may take the opportunity to line his pockets in the Chinese Super League if the opportunity presents itself again.
He added: “He should go if he feels [like it].
“Money now in modern football is high and if a Chinese or another rich team comes for him in the summer, he will have to go.
“Fans don’t attach themselves to the players, it’s the club, the club is the most important thing.
“But if I was him I would stay - nobody could put him in danger of losing his place at Chelsea.”