The Belgian forward continues to star for the Blues, with another inspirational showing put in on Saturday during a 4-0 victory over Brighton.
It is performances such as that which continue to see the 27-year-old mooted as a possible target for La Liga giants Real.
Sutton fears it could be only a matter of time before a big-money move is made, with Hazard’s impending departure set to leave those at Stamford Bridge with “a fair bit of business to do”.
The former Blues striker told the Daily Mail: “Chelsea fans should enjoy Eden Hazard while they can.
“They are a different team when he is playing.
“With Alvaro Morata out of form and Michy Batshuayi misfiring, Hazard has been Antonio Conte’s go-to man this season.
“He brings an unrivalled energy and sharpness to this team. As we saw against Norwich in the FA Cup last week, remove Hazard and Chelsea are far less threatening.
“Hazard is the biggest weapon in Chelsea’s armoury and to lose him this summer would leave them with a fair bit of business to do.
“Without him, Chelsea are still a good team. Despite a lack of strength in depth, they are third in the Premier League and still fighting on three fronts in the cups.
“But Hazard takes them to another level. Who can they rely on to do this when he is gone?”
Hazard could bring the exit talk to a close by committing to a new contract with Chelsea, but discussions regarding fresh terms are yet to see pen put to paper.
The Belgium international has not been aided by comments from family members either, with his father having revealed recently that an approach from Spain is being waited on.
“Yes. I talked with him [to tell him off], no, no worries.
“I'm focused with Chelsea. I've got two years contract left and I'm happy here. Like I said 10 times before, I want to finish this year and then we'll see. But I'm happy here.”