The 25-year-old Colombia international has been linked with a move to Manchester United and Chelsea has agreed a deal with Bundesliga champions.
“Signing James Rodríguez was our coach Carlo Ancelotti’s biggest wish,” said Bayern chairman Karl-Heinz Rummenigge.
“We’re delighted we’ve been able to complete this transfer.”
“James is a very versatile player. He’s a goalscorer himself, he sets up a lot of goals and on top of that he’s great from set-pieces,” continued Rummenigge.
“There’s no question that this transfer further increases the quality in our team.”