Osako, who featured at last year's World Cup in Russia, recently returned to Germany after scoring four goals for the Samurai Blue at the Asian Cup.
"We are very concerned that Yuya was fit to play in the Asian Cup semi-finals and final, but hasn't been able to play for us since returning to Bremen," the club's sporting director Frank Baumann said in a statement.
"Of course we are not pleased with the current situation."
Japan have been invited to play in the Copa America to be held in Brazil, starting June 14.
Bremen said they have informed the Japanese Football Association of their decision.
"Both the national association and Werder Bremen do not want one of Japan's most important players to suffer from burn out," added Baumann.
(Reporting by Jack Tarrant; editing by Sudipto Ganguly)
