Rodriguez, 27, has endured an uncertain future between his loan with Bayern and Real Madrid after falling out of favour with the Bundesliga giants this season.
The Colombia playmaker suffered niggling injuries over the past six months that has derailed his progress in Germany and reports in England suggested he could be on his way to the Premier League for the final six months of the season.
With the Gunners head coach Unai Emery admitting that his side can only afford to loan players in the mid-season transfer window, the chance of acquiring the Rodriguez appeared to be an appealing option.
However, the €75m signing of Los Blancos will remain for the final stretch of his 18-month loan, confirmed manager Niko Kovac.
"He wants to stay with FC Bayern, as he once said during a fan club visit, he is highly motivated and focused," Kovac said, as per The Mirror. "He can always help us with his quality."
Rodriguez has started just six games in the league this season before suffering a knee injury in November.