Alexander-Arnold has come under increasing scrutiny this season due to his defensive discipline, after being exposed in Liverpool's 1-0 defeat to Real Madrid on Wednesday night.
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The 24-year-old has struggled to replicate his form shown over the past few campaigns, and has often found himself caught out of position.
There has previously been suggestions he could be suited to centre-midfielder, the position he started out at as a youngster.
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Alexander-Arnold has previously been used in that position by England manager Gareth Southgate.
A move into the centre would see him follow in the footsteps of Phillip Lahm and Joshua Kimmich, who both started out at right-back before being moved into the midfield at Bayern Munich.
Jurgen Klopp, though, has claimed in the past he is not fond with the idea of moving the Liverpool star further forward.
Carragher believes a new approach is now needed, if Alexander-Arnold is to get back to his best.
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"That can't continue, what's happened this season with him and the number of times he has been lacking defensively," Carragher told told CBS.
"The problem they have got in that position is that they've never had real competition for Trent Alexander-Arnold.
"One of the reasons why is because he has been so good and you think 'Who do you buy and how much do you spend for someone who is not really going to play every week?'
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"Liverpool are at the point where they have got to go and buy a right-back. It's Jurgen Klopp decision in terms of how good that right-back is and how much he costs, but that lad needs serious competition.
"He has been playing for five years at the top level, with no rest, and has won absolutely everything and won more than I could ever dream of. I'm so proud of him - a local lad coming through the ranks - but he needs help.
"He needs to have a long think at the end of the season about where he is as a right-back, and I've always defended him because he's played in one of the best teams in the world.
"When you play for one of the best teams in the world, which Liverpool are, you do a lot more attacking than you do defending. We all know his defending is not good enough and I don't think it ever will be good enough."
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