Widespread reports emerged in England and Italy that the Blues had dismissed Conte after two seasons in charge.
The Premier League outfit is yet to make a statement on the speculation but is believed to be preparing to install former Napoli boss Maurizio Sarri as his successor.
In a post to Twitter, Fabregas wrote: "Thank you Mister for another Premier League title and FA Cup.
"It wasn't easy to convince you in the beginning but in the end I hope I did you proud! Good luck for the future."
Conte led Chelsea to the title in his first season in London before his tenure turned sour last term over rumoured frustration with the club's transfer policy.
Earlier in the day, Fabregas had 'liked' a tweet from a fan appearing to celebrate the Italian's departure, before removing the like and posting a reply that he had hit the button 'by mitake'.