Rodriguez is halfway through a two-year loan spell with the Bundesliga champions, and the club has the option to make the move permanent when the deal expires in June 2019.
However, the Colombia international is said to be interested in a return to the Santiago Bernabeu, and Real Madrid could look to negotiate with Bayern in an attempt to bring him back to Spain, per Marca's Jose Felix Diaz.
The 26-year-old joined Real Madrid from Monaco after playing a starring role at the 2014 FIFA World Cup, at which he scooped the Golden Boot. However, he failed to cement a regular spot in the starting lineup and left for Bayern in July 2017.
After a slow start to his time at the Allianz Arena, Rodriguez went on to play a key role and contributed eight goals and 13 assists in all competitions.
Bayern won the Bundesliga title for a sixth consecutive year. However, they were beaten in the DFB-Pokal final by Eintracht Frankfurt and knocked out of the UEFA Champions League by Real Madrid.
Rodriguez was part of Colombia's squad at the 2018 FIFA World Cup but saw his campaign interrupted by injury. Jose Pekerman's side were knocked out at the last-16 stage by England.
The Colombian played in a slightly deeper role under manager Jupp Heynckes last season as part of a three-man midfield, which enabled him to lay on assists for the Bayern attack.
WhoScored.com highlighted his effectiveness:
Niko Kovac has taken over as Bayern boss after Heynckes stepped down and will not want to lose one of his team's most important players. Rodriguez offered his view on Kovac in April, per the Bundesliga.
"He's a young coach, and he has done well at Eintracht Frankfurt," he said. "I hope he comes here with a big desire to win titles. That's what a big club always wants."
Rodriguez is a talented playmaker, and it's easy to see why Real Madrid might want him back. Los Blancos may have won the Champions League last season but could only finish third in La Liga. They have also since lost Cristiano Ronaldo to Juventus.
If they are to bring Rodriguez back, they will need to negotiate with Bayern, who have proved in the past they will not be bullied in the transfer market. The European champions may need Rodriguez to agitate for a move if they are to have any chance of seeing him return.