Martinez has been the subject of intense speculation this summer, with Bayern favourites for his signature.
Rummenigge told the club's official website: "We want to sign Javi Martinez for £31.6m, but there are still legal and tax problems.
"These problems could be solved very easily if Athletic Bilbao would accommodate us with a bit of help, but they are not doing so.
"We always said that we would never pay £31.6m for Javi Martinez, but he refrains from high wages to make this possible."
Bayern begin their attempts to win back the Bundesliga title against newly-promoted Greuther Furth on Saturday, August 25.