The Sun says Conte's outspoken comments have riled Chelsea's billionaire owner Roman Abramovich, who is set to wield the axe in the summer and make Enrique his 12th manager since taking charge in 2003.
Former Italy manager Conte, 48, wants to see out the final year of his Stamford Bridge contract, so will get a £9million pay-off.
Spaniard Enrique, 47, is out of work having left the Nou Camp last summer when his contract expired.
His unemployed status makes him an attractive option for Abramovich given he would not have to prise him away from another club or pay any compensation.
While Enrique is enjoying a year-long sabbatical, he is understood to be open to offers for next season.