Rodriguez, 27, joined Bayern on loan in 2017, and the German club have the option to sign him permanently for 42 million euros ($46.9m) when the two-year deal expires on June 30.
Yet according to Kicker, the recently crowned Bundesliga champions have decided against triggering the option-to-buy in Rodriguez’ contract.
“The Bayern bosses have agreed internally not to sign James,” wrote the magazine on Monday.
Rodriguez has scored 14 goals in 43 Bundesliga games for Bayern, but has had a difficult relationship with coach Niko Kovac this season.
The Colombian has also struggled to hold down a first-team place at Bayern, amid competition from the likes of Thomas Mueller, Leon Goretzka and Thiago Alcantara.
With a return to Real Madrid considered unlikely, Kicker reported that Rodriguez is on the verge of a move to Serie A club Napoli, where he would be reunited with former Bayern and Madrid coach Carlo Ancelotti.
The magazine also claimed that Bayern were yet to make a decision over the future of 21-year-old Portuguese midfielder Renato Sanches, who has repeatedly demanded more game time in recent weeks.