The Portuguese left the club in December with the Blues struggling at the wrong end of the Premier League after losing nine out of 16 games.
The 2015 champions are now 10th -- 16 points outside the Champions League places -- and Fabregas said he and his Chelsea team-mates "let him down."
"I have huge respect for him -- we still keep in touch right now," the former Arsenal and Barcelona player said on Sky Sports.
"The biggest problem was he trusted us too much, gave us more holiday because we were champions and we let him down.
"That was the main reason he had to go -- and for that myself and the team feel bad for it."
Fabregas, who spent eight years at Arsenal, also said he wanted Leicester to win the title, with the Foxes still to travel to Chelsea in their final game of the season on May 15.
Asked, ahead of Tottenham's meeting with Stoke, if he felt there would be another twist in the title race, he said: "Hopefully not, I don't want Spurs to win it to be honest.
"For what they've done through the season I'd love Leicester to win the Premier League."
Fabregas will next year compete against his former manager at Barcelona when Pep Guardiola takes over at Manchester City.
The midfielder said he loved playing for Guardiola, who was not afraid to change things up despite the club's success when he came on.
"What [Guardiola] wanted to do is change formation," Fabregas said. "I think after three years playing at a fantastic level and winning many things, he felt the other teams were catching the way of playing against Barcelona on a 4-3-3," said Fabregas on Sky Sports.
"He wanted to play a different system with three at the back, four in midfield plus Lionel Messi dropping deep - he wanted to dominate midfield more than anything -- and to have to wide players always open and threatening to go in behind defenders.
"Yes, [I enjoyed it] because I had a lot of freedom. They tell you play wherever you feel you are dangerous and I love this kind of thing, because it's like a No. 10, but with the mentality of the goal scorer."
Fabregas said he also expects Guardiola to similarly shake up the Premier League when he arrives in the fall.
"You always want to play against the best, 100 percent Pep will bring new things to the Premier League, to Man City," he added.
"They will be a force next year, they are already this season, you see they are in the semifinals of Champions League and fighting to be in t
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