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The Reds lost 2-0 at Burnley, as they were punished for a series of mistakes, including losing possession for Sam Vokes' second-minute opener as they tried to play out from the back.
"There are real issues," Murphy said on Match of the Day. "Are they good enough to play from the back all the time?
"They're not Barcelona. They have to mix the game up better."
Former Manchester United defender Phil Neville added: "When Jurgen Klopp came to the Premier League we thought he would be a direct coach, get the ball forward and play fast, attacking football
"To play this system well you have to have your centre-backs control the game. Against Burnley they were poor."
Klopp, in his first full campaign in charge of Liverpool, said: "The season is still pretty young. It doesn't look like we are 100% fine-tuned. But it's only the start of the season and I can see a lot of good things.
"The lesson here is don't give the ball away like we did twice today. We will work on it 100%. We need to have a plan for deep-defending teams. We have a few things to do - that is clear."
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Liverpool have "real defensive issues" that need to be addressed, says former midfielder Danny Murphy.