Conte has come under pressure following two heavy defeats in less than a week for his side.
Seven months after helping them to their 4th Premier League title, Chelsea sack Jose Mourinho after a string of poor results, the embarrassing 2-1 defeat to Leicester City being the last straw.
After hitting new low's with losses against Bournemouth and Watford in recent weeks, Chelsea may struggle to qualify in the Champions League for next season if they are unable to secure a top 4 spot in the Premier League table.
Chelsea have reportedly chose to open talks with the Spanish manager who is likely to take over at the Stamford Bridge before the end of the season.
The caricature of Abramovich as a fickle owner who turns against his managers on a whim expired with his sacking of Carlo Ancelotti, the one coach who was dismissed too soon and the one departure the club regret.
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However, Barcelona's 22nd game unbeaten broke their own club record and kept the possibility of a broken league record on track. After eliminating Espanyol from the Copa del Rey, Barcelona gets another crack at its city rival in league play on Sunday.
The Mail say the players are unhappy with Conte's training methods and apparently his mood swings.
Enrique has been out of work since leaving the Catalan giants at the end of last season. Camp Nou legend Xavi had to intermediate a problem between the manager and some players in the 2014-15 campaign, although it did end with the club claiming a treble.
But, considering various reports in England have claimed that Chelsea have not identified Conte's successor, it's probably worth taking this particular story with a pinch of salt.
Chelsea face West Brom at Stamford Bridge on Monday evening, which may not seem like the most intimidating fixture but does have ominous overtones.
In France, Le10 Sport have gone a step further, declaring that Enrique will look to make Barcelona's former Liverpool striker Luis Suarez his first signing if he takes over at Chelsea. If this is true, it would explain why Conte seems very relaxed about his future.