The Blaugrana are not in a position to meet the versatile midfielder's buyout clause but it is believed that they now view the as a better option than AC Milan's Thiago Silva
According to El Mundo Deportivo, the Blaugrana are attracted to the player's versatility, with the 23-year-old just as comfortable at centre-half as he is in midfield.
Barca also view Martinez as the ideal replacement for Seydou Keita, who is expected to depart Camp Nou this summer.
The Club World Cup champions have no intention of meeting the €40m buyout clause in the Estella native's contract and are even reluctant to hand over €10m less than that.
However, El Mundo Deportivo reports that that there is a growing feeling within Camp Nou that Martinez represents better value for money than AC Milan centre-half Thiago Silva in light of his youth and aforementioned versatility.
The Blaugrana are therefore hoping that Athletic might be willing to lower their asking price or at least accept a player-plus-cash deal, perhaps involving Barca B team goalkeeper Oier Olazabal.
However, Barca are aware that they face stiff competition for Martinez's services from both Manchester United and Manchester City and have no interest in getting involved in a bidding war with the Premier League powerhouses.