Fabregas was tearful as he applauded all sides of Stamford Bridge following Saturday's 2-0 FA Cup third-round win over Nottingham Forest, during which he had a first-half penalty saved.
The 31-year-old's contract expires this summer and he has been linked with a January move to Monaco and a reunion with former Arsenal team-mate Thierry Henry.
David Luiz revealed Fabregas addressed the dressing room afterwards.
"He did a speech and then after that we all said thank you for him, and hugged him," said Luiz, who declined to reveal what Fabregas said.
Any move is still to be completed, but Fabregas' emotional reaction after his 501st appearance in English football suggested it was goodbye.
Luiz added: "It's always difficult to lose players, especially like him, because, like I say, he's a world-class player but also I understand it's part of the process, and if it's better for him and better for the club, we have to understand that.
"One of the best players playing in England was him, everyone knows that.
"(It is) sad for everybody when you lose this kind of player, this kind of person, but I think everybody has to stand up and clap their hands for him - because he deserves it."
Eden Hazard also paid tribute to Fabregas, who joined Chelsea from Barcelona in June 2014, two years after Hazard, and has won two Premier League titles, plus the 2015 League Cup and 2018 FA Cup, during his spell at Stamford Bridge.
"In five years, I can say that I was lucky to play with him," Hazard said.
"Since the beginning when he came, we saw it the first year, we won the double together.
"Top guy on the pitch, off the pitch, a great friend of mine.
"Just admiration about this guy. I hope he will have a good future. It's a bit hard now, but he's still very good and he will enjoy."
Another player who has been linked with a move away is Callum Hudson-Odoi (above), who set up both goals for Alvaro Morata against Forest.
The 18-year-old winger, whose contract expires in June 2020, is reportedly a target for Bayern Munich.
Hudson-Odoi is eager for further opportunities and knows he could be given them, due to Chelsea's injury situation.
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Striker Olivier Giroud (ankle) and winger Pedro (hamstring) could come back into contention for Tuesday's Carabao Cup semi-final first leg with Tottenham.
But Ruben Loftus-Cheek appeared to aggravate a back injury against Forest in a game Willian missed with a hamstring problem sustained against Southampton last Wednesday.
Hudson-Odoi told Chelsea TV: "I've just got to be patient and keep working hard, no matter what. I've got to keep pushing in training.
"The injuries are unfortunate. For me I've got to keep working hard and hopefully my chances will come."