Express Sport understands Chelsea are ready to axe Conte should the Blues slip to defeat against Watford at Vicarage Road tomorrow (8pm).
Former Barcelona boss Enrique is in pole position to take over, with the Spaniard currently on a sabbatical after leaving the Catalan giants at the end of last season.
Conte’s position at Chelsea has grown weak due to his regular criticism of the club’s transfer policy.
And Spanish website Don Balon say Enrique is seeking guarantees from the west London club, and wants them to prove they are capable of signing big players by landing Dortmund star Reus.
The Germany international’s injury record has seen his stock fall in recent years.