The Reds lost 3-1 to La Liga giants Real Madrid in the Champions League final in Kiev last weekend.
Karius made two mistakes to allow Karim Benzema and Gareth Bale to find the net in a resounding win for the defending champions.
The Liverpool FC goalkeeper has made improvements to his game over the past few months after replacing Simon Mignolet as number one.
Former Borussia Dortmund head coach Klopp stuck with Karius despite an uncertain start to his Liverpool FC career in the 2016-17 campaign.
The Reds have been linked with a potential move to sign AS Roma goalkeeper Alisson Becker or Altetico Madrid shot stopper Jan Oblak.
Former Liverpool FC defender Lawrenson believes the Reds need to offload Karius despite Klopp’s temptation to stick with the German goalkeeper.
“Even if Jurgen Klopp backs him, if he does I will find it strange, the supporters – every time he plays – will be waiting for him to make another mistake,” Lawrenson told BBC Radio 5 Live.
“Therein lies the reason they’ve got to say sorry but no thanks.
Liverpool FC are in fourth place in the Premier League table and finished five points ahead of Chelsea FC in the Premier League table.
The Reds will look to challenge Manchester City for the Premier League title next term.
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