Klinsmann has failed to impress in his first season at the German champions, his side crashing out of the German Cup and lying fourth in the Bundesliga, three points off the top with eight matches left.
Their capitulatoin against a dazzling Barcelona team on Wednesday rubbed further salt in the wounds of Bayern fans who have been unhappy with Klinsmann's style of play.
Consistency and last year's rock-solid defence have been sorely missing this season despite Klinsmann's pre-season pledge to "make every player better."
Bayern were hammered 5-1 by league leaders wolfsburg last week before their flop against Barcelona.
However, Club chiefs Karl-Heinz Rummenigge and Uli Hoeness had some sharp words to say about the team's results but insisted there would be no heat-of-the-moment decisions over Klinsmann
"One should not take any decisions today," Rummenigge told reporters. "It was a disgrace but it is best if you sleep and then deal with these issues rationally."
Asked whether Klinsmann would be on the bench against Eintracht Frankfurt on Saturday, Rummenigge added: "This is the starting point. But I do not want to rate the coach after every match."
While Rummenigge, the club's general director, has backed Klinsmann throughout the season but club president Franz Beckenbauer has been less supportive.
"This is a catastrophe," Beckenbauer told Premiere Television after the 4-0 defeat against Barcelona. "This is schoolboy stuff. The first half was the worst I have ever seen Bayern play."
With Bayern now almost certainly out of the Champions League, Saturday's league match against Eintracht Frankfurt is crucial as they seek to salvage their season.
Failure to win in Frankurt could severely dent their title hopes and possibly end Klinsmann's brief reign in charge.
