With his current deal due to expire in June, the Scot has shelved any possible negotiations until the summer, stating that he wants to gauge the Toffees' progress between now and the end of the campaign.
That has led to intense speculation that his 11-year reign with the club could be coming to an end.
According to The Mirror, Martinez, who was interviewed for the Liverpool job last summer, is the leading candidate to succeed him at Goodison Park.
The Spaniard is also believed to be wanted by Chelsea, as Rafael Benitez's temporary spell in charge looks certain to come to an end.
Martinez took over at the DW Stadium in 2009 after a successful spell at Swansea City, whom he led to promotion from League One in 2008.