According to Marca's Jose Felix Diaz, talks between the Spaniard and the Blues broke down after initial optimism because of Enrique's personality and perceived "arrogance," while Arsenal did not like his demands about the building of the squad.
Per Diaz, a Chelsea source said: "There was no real feeling with him at any time."
Chelsea are preparing to replace Antonio Conte in the top job, with the Italian looking set to depart the club following Saturday's FA Cup final at Wembley Stadium against Manchester United.
Maurizio Sarri of Napoli previously looked to be the front-runner, but recent reports have indicated he is in talks to sign a new deal at the Serie A club, per Sky Sports News.
The Gunners, meanwhile, face the mammoth task of finding a successor for Arsene Wenger, who has stepped down after nearly 22 years in charge.
According to David Ornstein of BBC Sport, former Arsenal captain Mikel Arteta is the likely appointment.
Enrique has been unattached since departing Barca last summer having won two La Liga titles and a UEFA Champions League, as well as the Copa del Rey three times, in his three-year spell at the helm.
With a CV like that, it is no surprise Arsenal and Chelsea were initially interested, but clearly neither club felt Enrique was right for the job.