The Italian coach has come under increasing pressure following back-to-back defeats in the Premier League.
Enrique emerged as a possible contender to replace Conte a few days ago, and after yesterday's result, the speculation has intensified.
According to Diario Sport, the former Juventus boss is expected to be sacked straight away and Enrique is first in line to become Chelsea's new manager.
The report adds that ex-Blues full-back Juliano Belletti, who also played for Barcelona in his career, could become the new sporting director at Stamford Bridge.
Enrique has been out of work since leaving the Catalan giants at the end of last season.