Having finished third in the German Bundesliga last season after being huge favourites, Bayern are eager to avoid a repeat next term and have already made drastic changes at the club.
Jupp Heynckes has been brought in as Bayern's new coach, after Louis van Gaal was sacked near the end of last season, while Germany number one Manuel Neuer and Brazilian full-back Rafinha have arrived as summer signings.
However, Bayern are unlikely to be done with their spending this summer, and have long held an admiration for Boateng, who has struggled to establish himself at City.
While the Bavarian outfit have been rebuffed with their initial offer, Bild believes they are ready to up their bid to €16million, which they hope City will view as an acceptable figure.
Despite his progress stalling at Eastlands last term, Boateng is still regarded as one of German football's top prospects, and is still a regular in German coach Joachim Low's team when fit.