The Blues are believed to be on the verge of parting ways with Antonio Conte, who failed to secure Champions League football last term and often clashed with the club hierarchy.
Maurizio Sarri was thought to be Chelsea's primary target, but negotiations have reportedly broken down due to Napoli seeking compensation for the Italian.
According to Spanish outlet Sport, the Blues have turned their attentions to securing a deal for Enrique, who has been out of work since taking a sabbatical after parting company with Barcelona at the end of last season.
The Spaniard was thought to be in the running to succeed Arsene Wenger at Arsenal and had been championed by director Raul Sanllehi, but former Paris Saint-Germain coach Unai Emery eventually took the reins at the Emirates Stadium.
Enrique has also managed Barcelona B, Celta Vigo and Roma, and was a hugely successful player, making upwards of 350 appearances for Barca and Real Madrid.