A disappointing season saw the defending Premier League champions finish fifth in the table, five points off the top four and 30 behind new champions Manchester City.
Chelsea did end their campaign with silverware by beating Manchester United in the FA Cup final at Wembley on Saturday, but Conte's future is understood to have been decided before then.
The Express reports that official confirmation of the Italian's departure is expected in the coming days, with former Barcelona boss Luis Enrique thought to be at the top of the list of potential successors.
Enrique has been out of work since leaving Barcelona after a trophy-laden spell last summer, while Monaco boss Leonardo Jardim and Napoli's Maurizio Sarri are also thought to be on Chelsea's wishlist.
Conte, who will be due a £9m payout for the final year of his contract if he is sacked, won the Premier League title and FA Cup during his time in English football.