The 25-year-old Colombia international made 22 La Liga appearances for Real last season, scoring eight goals, and featured six times in the UEFA Champions League.
"We are very happy to have completed this transfer," Bayern Executive Board Chairman Karl-Heinz Rummenigge said in a statement. "Signing James Rodriguez was the big wish of our coach Carlo (Ancelotti) after they worked successfully together in Madrid."
Rodriguez moved to Real Madrid from Monaco for a reported 60 million euros (53.11 million pounds) in 2014 after winning the Golden Boot as the leading scorer at the FIFA World Cup. He was also named in the team of the tournament as Colombia reached the quarter-finals.
His ability to score spectacular goals made him an ideal addition to Real's "Galacticos" project, but he struggled to nail down a regular starting place at the Bernabeu.
Real confirmed that Bayern could make the move permanent at the end of the loan period.
"Real Madrid and Bayern Munich have agreed to the transfer of player James Rodriguez for the next two seasons, pending a medical examination," the Spanish club said on its website.
"Bayern Munich reserves the right to purchase the player at the end of the transfer period."
Rodriguez will join up with Bayern's squad immediately and is expected to travel with the team for their 12-day pre-season tour in China and Singapore.
The Colombian is the German club's sixth signing of the close season, following the arrivals of wingers Serge Gnabry and Kingsley Coman, midfielders Corentin Tolisso and Sebastian Rudy, and centre back Niklas Suele.
(Reporting by Simon Jennings in Bengaluru, editing by Christian Radnedge and Pritha Sarkar)
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