Sane joined the Bavarians from Manchester City last summer but has so far produced a number of inconsistent displays at the Allianz Arena.
The latest came in Bayern Munich’s 3-2 victory over FC Cologne in the Bundesliga on Sunday, when Sane was hooked off at half-time and jeered by the home fans.
Rummenigge has sympathized with the Germany international and asked him to take inspiration from former Dutch winger Robben, who bounced back from missing a crucial penalty in the 2012 Champions League final against Chelsea to become an all-time Bayern Munich great.
"At first I didn't like it, I feel sorry for him [Sane]," Rummenigge said on Bild's Reif ist live podcast about what happened on Sunday.
"He tries but has no confidence. He didn't play a good game. With his transfer fee and salary, the criticism of the fans is slowly coming up.
"I remembered 2012 in the stadium. There was the famous home final [in the Champions League]. It reminded me of the sad Arjen Robben. He missed a penalty in the final.
"Arjen was almost ready to ask for his release. But we said: 'No, we won't do that.' He was an important player for Bayern Munich. We supported him.
"Jupp Heynckes built him up as a coach. He was extremely ambitious. The result: a year later he was the match-winner in the Champions League final against Dortmund."
Sane featured 32 times and scored six goals as Bayern Munich won the Bundesliga last season.
