Marca says the Bayern Munich midfielder has his heart set on a return to San Mames during this summer transfer window.
He is expected to accept a two-year deal, with the option of a further year, in order to re-join his former club.
What remains to be finalised is the transfer fee, as Martinez still has one year left on his contract at the Allianz Arena.
Bayern started negotiations with a 10 million euro price tag but are now willing to accept a lower amount.
Although Martinez expects Bayern to facilitate an exit this summer because of the length of his contract, he knows the German club won't gift him his return to Bilbao.