Rodriguez, who turns 26 on Wednesday, signed for Real following good performances for Colombia at the 2014 World Cup.
However he has struggled for regular playing time and has slipped behind the likes of Marco Asensio and Isco in the pecking order.
The Bayern statement says that there is an option to purchase during the loan but they do not state the fee.
Chairman Karl-Heinz Rummenigge said “We are very happy that we were able to implement this transfer. The commitment of James Rodríguez was the great wish of our coach, Carlo Ancelotti, after both had already successfully worked together in Madrid,
“James is a versatile player. It is self-dangerous, prepares many hits and shoots great standards. With this transfer, we have been able to raise the quality of our team without question. “