The deal would eclipse the €35m that Bayern paid Stuttgart for Mario Gomez in 2009.
"The transfer is in a good position, but not done yet," Bayern coach Jupp Heynckes said after the pre-season friendly against Seligenporten. "We have to be patient for a bit longer."
German newspaper Bild reported that a €40m fee had been agreed for the 23-year-old.
Bayern sporting director Matthias Sammer has prasied the Spain international, whom Bayern have coveted all summer.
"His stature and charisma alone make him a character at the centre of play," he said. "He has enormous strategic skills, great passing abilities, he is physically tough and has a powerful heading ability."
Rumours of a deal grew after Martinez was substituted at half-time in Athletic's pre-season friendly against Basque minnows Barakaldo on Wednesday and he drove away before the end of the match.
His agent Antonio Sanz has denied the reports of a fee being agreed, saying Athletic are not willing to sell and that his player will remain at the San Mames.
"This is a complete lie," he said. "The clubs are not negotiating, as Bilbao is not willing to negotiate.
"I do not believe Bayern have put €40m on the table. Javi Martinez will play for Athletic Bilbao next season."
Bayern have already signed Mario Mandzukic for €12m, Xerdan Shaqiri for €9m and Dante for €4.7m after a promising season ended without a trophy for a second successive season following failure in the Bundesliga, the German Cup final and Champions League final.