And now the Mail on Sunday reckon he is the frontrunner.
According to the report, Enrique is the obvious target for Chelsea because their two other candidates - Juventus boss Massimiliano Allegri and Atletico Madrid's Diego Simeone - both have two years remaining on their contracts.
It would cost Chelsea a lot of money to take either man away from their current clubs.
Chelsea do not want another expensive managerial swap after sacking Jose Mourinho in December 2015 cost them a fortune.
Conte earns £6.5million a year at Chelsea and his contract expires in 2019.
Conte: Whole Chelsea team need to improve on goal-scoring
But the manager and the club may happily go their separate ways this summer.
As for Enrique, he has been out of football for a year after leaving Barcelona at the end of last season.
The former Celta Vigo and Roma boss won two La Liga titles with Barcelona, the Champions League and three Copa del Rey trophies.
The Mail on Sunday say he is open to a deal and is the best available option.
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