Rodriguez did not play in the German Super Cup or the first round of the Deutsche Pokal due to an ankle injury, but since his return to fitness, he has been used only as a substitute.
The Gunners were said to be interested in the summer when it first emerged that the 27-year-old could be allowed to leave the Allianz Arena.
Don Balon believe the playmaker could cut his loan spell short at Bayern and move in the January transfer window. The German club do have the option to buy Rodriguez for €42m (£38m) at the end of the season, but the Spanish website say Real could now look to sell the player for €50m (£45m) to another club.
Take this rumour with a large pinch of salt. Don Balon are notoriously inaccurate with their transfer stories, but if Rodriguez is available, there would certainly be a number of clubs keen to take him across Europe.