Big Sam, who has struggled to win over the fans, is under pressure despite a season in which the club have flirted with European qualification.
It means the Hammers board are looking elsewhere and believe they would be able to lure Martinez (above) away from Goodison Park.
Ironically they have already sounded out former Everton manager David Moyes, who has moved to Spain with Real Sociedad but he has indicated for now he is staying put in La Liga.