Many fans among the 33,000 crowd at Tuesday's friendly between Bayern and Netherlands in Munich vented their frustration at losing out on the biggest European club prize, whistling and jeering the Dutch international, who played only for Netherlands, when he had possession.
Robben's spot-kick in extra time on Saturday was saved by Chelsea keeper Petr Cech. The English club went on to become European champions after winning a penalty shootout at Bayern's own Allianz Arena. Robben did not take a penalty in the game-deciding shootout.
"Just like the other players, fans and friends of Bayern, Arjen Robben is deeply disappointed and affected by the undeserved defeat on penalties against Chelsea," Bayern CEO Karl-Heinz Rummenigge in a statement.
"But it is the style of Bayern and their fans to give the one who has fallen down the hand and help him. This morning we have received a lot of reactions from Bayern fans who apologise on behalf of those who whistled.
"Bayern wish Arjen Robben a good and successful Euro championship and look forward to having him back in the Bayern group in July."
Robben, who earlier this month extended his Bayern contract by two years to 2015, has had an uneasy relationship this season with the club's fans, some accusing the Dutchman of being too selfish on the pitch.
He was briefly benched earlier this year by coach Jupp Heynckes and a penalty miss against champions Borussia Dortmund in a key league game in April did little to appease Bayern fans.
Netherlands coach Bert van Marwijk called Bayern fans' behaviour "a disgrace" on Tuesday, while former team mate Mark van Bommel said Robben should consider his Bayern future.