Off contract in June, Alexander-Arnold is being linked with Real Madrid.
Last week, Alexander-Arnold gave an interview declaring he wants to win the Ballon d'Or, which Carragher believes is a clear sign he wants a move to Spain.
Carragher told Sky Sports: "I don't care, he can try to win the Ballon d'Or, nothing wrong with ambition but my first thought when I read that was ’you’re not gonna win the Ballon d’Or as a right back for Liverpool.
"Ballon d'Or winners normally play for Real Madrid or Barcelona. The vast majority, the players in the world go. If that’s his ultimate ambition - I think he was asked about winning the World Cup, Champions League or Ballon d’Or - and he chose the Ballon d’Or - which I think was a bit strange or bizarre because I’d always pick World Cup or Champions League, it’s a team game.
"But yeah, he’s got massive ambition and why not? You only get one career, he is a unique full-back, he is absolutely amazing. He gets Liverpool back into the game today with the pass but the first thing that came to me when I read that interview was that makes me think he’s going to Real Madrid. I actually think Virgil van Dijk and Mo Salah will stay, I do believe that. The longer it goes on with Trent, the more I do feel he will move on."