Liverpool ended a four-game winless run by prevailing 3-2 at Leicester City on Saturday, but they have conceded 10 goals in three consecutive Premier League away games for the first time in the competition's 25-year history.
Klopp's men have kept only two clean sheets in 10 matches this season and he admitted after last weekend's 2-0 loss at Leicester in the League Cup that he was "really, really sick" of shipping soft goals.
Coutinho committed to Liverpool cause
Held to a 2-2 draw by Sevilla in their Champions League opener two weeks ago, Liverpool have failed to keep a clean sheet in 11 of their last 12 matches in the competition.
But for all their defensive woes, which are symbolised by recurrent failings at set-pieces, Klopp has been reluctant to admit that he has a major problem on his hands.
Asked about his team's difficulties at the back prior to their most recent game at Leicester, he protested: "It sounds like we have no points and are completely the worst team in the league."
Klopp eyes Leicester scalp to end disastrous September
If Klopp has been spared more trenchant criticism, it is because his team, when on song, are capable of playing attacking football of scintillating quality.
Liverpool crushed Arsenal 4-0 at the end of August and in Sadio Mane, Roberto Firmino, Philippe Coutinho and new signing Mohamed Salah they possess some of the most talented forwards in the Premier League.
The suspicion, which Klopp does not exactly deny, is that as long as his team are scoring goals at one end of the pitch, he will not lose too much sleep about what is happening in his own penalty area.
"If you would ask me: 'Do you want your team perfect in defending set-pieces, but don't know how to play football, or would you prefer the other way round?' I still would take this (second) one," he said.
Liverpool's quest for greater defensive solidity has not been helped by the unpredictable form of goalkeeper Simon Mignolet or by Klopp's own selection choices.
The 5-0 capitulation at Manchester City earlier this month came after he made the questionable choice to start Ragnar Klavan instead of usual first-choice centre-back Dejan Lovren, who remained on the bench.
The Liverpool manager also regularly chops and changes his defence, notably rotating his full-backs between games, which has made it impossible to achieve any kind of continuity.
Matters were not helped by Liverpool's frustrating transfer window, during which they missed out on prime defensive target Virgil van Dijk.
With Liverpool having failed to prise the Dutch centre-back from Southampton, former captain Phil Thompson believes it was a mistake to allow Mamadou Sakho to join Crystal Palace after the Frenchman fell out with Klopp.
"Sakho was our best defender," Thompson told Sky Sports last week. "I know there are fallings out at football clubs all the time, but somewhere along the line, you can't cut off your nose to spite your face. Klopp should have made up with the boy. He should have banned him, but then brought him back into the fold. They lost out on Virgil van Dijk for whatever reason, but when it happened, they should have kept Sakho."
Overall, Thompson feels there "shouldn't be any debate" about Klopp's position and says the German has clearly taken Liverpool forwards since succeeding Brendan Rodgers in October 2015.
But while Liverpool's attacking football will continue to nourish hopes of a return to prominence at home and abroad, their defensive issues may prove a fatal flaw.
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