The Blues have been in woeful form this season in the Premier League and slumped further toward the relegation zone on Sunday with an embarrassing 4-0 defeat at the hands of Merseyside rivals Liverpool.
The Anfield humiliation has put further pressure on the shoulders of the Spanish coach, whose team are now just two points better off than lowly Crystal Palace in 16th place.
According to the Mirror, new major shareholder Farhad Mashiri is eager to employ a manager who will enable the club to become a major force.
Current chairman Bill Kenright will reportedly remain in control of the Goodison Park club for the next two years before Mashiri - who is less than bowled over by Martinez’s showing this season - will take over.
Martinez’s side face Manchester United in the semi-final of the FA Cup on Saturday, with the Mirror suggesting the former Wigan Athletic boss must at least secure a cup triumph to persuade his future employer he is the correct man for the job.