The 27 year-old Italian striker holds a good record in Serie A, but looks set to be limited to cameo appearances following the summer arrivals of Carlos Tevez and Fernando Llorente.
Since his arrival at the Turin giants from Cagliari in 2011, the Lombardy-born Italian has impressed, scoring 27 goals in 69 competitive appearances for 'The Old Lady' of Italian football.
Manager Antonio Conte is now prepared to sacrifice the forward following his summer additions, allowing himself to strengthen in alternative departments.
Despite adding former Wigan striker Arouna Kone to his attacking ranks, it is understood that Martinez continues to view the forward positions as ones which still require addressing.
Midfielder Marouane Fellaini was the club's top scorer last season with 12 goals, amid poor showings from Nikica Jelavic and Victor Anichebe.
Matri would cost Everton in the region of £12m to buy, with the player seemingly interested in the prospect of a switch to the Premier League and to Goodison Park.
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