Klinsmann's side have conceded nine goals in their last two games
Furious Bayern Munich president Franz Beckenbauer condemned the first-half performance against Barcelona as the worst in the club's history.
Barcelona humiliated Bayern, scoring all of their goals in a 4-0 win before half-time of their Champions League quarter-final first leg.
"What I saw in the first half is, without doubt, the worst football in Bayern's history," said Beckenbauer.
"It was a demonstration, almost a humiliation. They gave us a lesson."
Inspired by Leo Messi, Barcelona swept aside Bayern to take complete control of the tie ahead of the second leg at the Allianz Arena on Tuesday.
Messi scored twice and was unfortunate not to win a penalty, with Thierry Henry and Samuel Eto'o also taking advantage of some shambolic defending by the visitors.
We had a very good performance but it's only the first game of the quarters... we're not in the semi-finals yet
Barcelona boss Pep Guardiola
The result heaps further pressure on Bayern manager Jurgen Klinsmann, who saw his side thrashed 5-1 by Wolfsburg in the Bundesliga at the weekend.
Chairman Karl-Heinz Rummenigge refused to comment on Klinsmann's future.
"What I want to say is that we must now concentrate on our next match against Frankfurt on Saturday," Rummenigge said.
"We have no more hope in the Champions League, that is obvious. Now we have to concentrate on the Bundesliga and at least make sure of our place in this competition (for next season)."
Klinsmann praised Barca, but said injuries to Phillip Lahm and Lucio as well as Daniel van Buyten's unavailability had hampered his side.
"It is a big disappointment," he said. "Barca played very well and you have to congratulate them. Messi is simply a wonderful player.
"It was difficult to compensate for the injuries and the two early goals we conceded hurt us a lot."
Barca boss Pep Guardiola was understandably delighted with his side's efforts, although he cautioned against complacency.
"We had a very good performance but it's only the first game of the quarters," he said. "We're not in the semi-finals yet.
"Our midfield got out well and moved the ball around efficiently and quickly and that is the best way to defend.
"In general we allowed them little room to breathe."
Guardiola spent most of the game watching from the stands after losing his cool over Messi's failure to win a penalty in the first half.
The Argentine appeared to be tripped by Christian Lell but referee Howard Webb turned down the penalty appeal and showed Messi a yellow card.
"I wasn't able to control myself, I'm like that. The penalty was clear," said Guardiola. "What bothers me more is the card."
While Messi was the game's outstanding player, Henry showed flashes of his old Arsenal form too, in particular when setting up the third goal for his Argentine team-mate.
The Frenchman said hard work was even more crucial to Barca's current form than flair.
"It's about how we fight for each other," he told BBC 5 Live.
"When people talk about Barcelona they always talk about the way we play and how we pass the ball.
"But I'm not sure if you noticed but we ran, we worked hard to get the ball back, we put pressure on their team and I think that's the most important thing."
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