The 28-year-old was playing for Holland in a friendly match against his club side, and was booed each time he touched the ball following his 75th-minute arrival on the pitch.
Oranje were beaten 3-2 by Bayern just three days after the Bundesliga club lost out to Chelsea in the Champions League final.
Robben missed an extra-time penalty in that match, which was also held at the Allianz Arena, and in a statement today [Wednesday] his club condemned the jeers.
"Some fans were clearly disappointed that Arjen Robben was playing in the Holland team and not in a Bayern Munich shirt just three days after the lost Champions League final," said chairman Karl-Heinz Rummenigge.
"This, however, does not give anybody the right to boo one of our players.
"Arjen Robben is, like all of the players, fans and friends of Bayern Munich, disappointed and deeply hurt by the unlucky and undeserved defeat in a penalty shoot-out in the Champions League final against Chelsea.
"But the style of FC Bayern and our fans is to reach out and help those who are lying on the ground, like the majority of the 33,000 fans did by chanting Arjen Robben's name.
"We have received lots of reaction from our fans today, who want to apologise to Arjen Robben on behalf of those who whistled."
The incident also drew condemnation from Holland captain Mark van Bommel, who urged Robben to reconsider his Bayern future despite recently signing a contract extension until 2015.
"It's absolutely outrageous how the fans treated Arjen Robben," Van Bommel told Voetbal International.
"You only have to look at what he has accomplished there. He scored a penalty against Real Madrid in the (Champions League) semi-finals.
"People seem to have forgotten that. They should be pleased to have such a player.
"A large part of the German crowd booed their own player. That cannot happen.
"If I was Arjen, I would start thinking about whether I still want to play football there, even if my contract has just been renewed.
"Arjen is very strong mentally, and comes back stronger each time after every setback. I don't know what Arjen is going to do but this is a disgrace."
Holland coach Bert van Marwijk also said he would understand if Robben wants to leave Bayern after being booed by their fans.
"It is a disgrace the way they treated Arjen," said Van Marwijk. "I can understand when a player considers whether he should leave or not in such a situation.
"That is not funny anymore, that is serious."
The game was arranged by means of compensation to the Bundesliga club after the Dutch forward returned from the 2010 World Cup with an injury.