Reports from Mundo Deportivo claim the Catalans have switched their attention to the Basque defender after losing out to Chelsea in the race for Jules Kounde.
Martinez offers a more experienced option than Kounde, with over a decade of top-flight games under his belt, since breaking through the youth ranks at Real Sociedad.
The 31-year-old is a vital player for Athletic Club, and new boss Valverde is aware of his importance, and he is not welcoming bids for him this month.
“He hasn’t said anything to me at all. We are all very calm about it”, as per reports from Marca.
“We have players who are very committed, you see him in every training session and every game, he’s a very important player for us.”
Barcelona have brought in Danish international Andreas Christensen, to boost their defence ahead of the new campaign, and Pedri has tipped his international teammate Martinez to make the move before the start of the new season.
La Blaugrana will step up their interest, with less than 12 months left on his current contract, and he could be available for €22m.