In his lengthy career in Germany, Pizarro has made 327 appearances for Bayern as well as 296 for current club Werder Bremen.
A well-loved figure at both clubs, he is currently on his fourth spell at Bremen, and is expected to end his playing career either this summer or at the end of next season.
Pizarro is the second highest scoring foreigner in Bundesliga history with 195 goals in 468 league appearances - only Bayern's Robert Lewandowski, currently on 201 goals, has scored more in Germany's top flight.
In an interview with Bremen website "Mein Werder", Bayern CEO Rummenigge admitted that he hopes to lure Pizarro back to Bayern once the Peruvian hangs up his boots.
"We have agreed with Claudio that he can come back and take on a role as a club ambassador when he actually does stop playing," said Rummenigge.
"As far as I know, he still has a house in Munich and his family always felt very at home here."
Several former Bayern players, including Bixente Lizarazu and Giovane Elber, are active as official Bayern ambassadors, a role which involves representing the club and the brand at publicity events across the world.
For the time being, however, Pizarro is focused on beating Bayern in the league and cup double bill over the coming week.
Bayern host Bremen in the league on Saturday before the two teams meet in a German Cup semi-final next Wednesday.
"It would be an amazing feeling for me to win these games, though I would feel Bayern's pain a bit," said Pizarro this week.
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