The full-back is said to be closing in on a move to Real Madrid with his contract at Anfield set to expire at the end of the season but the Sky Sports pundit is saddened that the player is walking away from the chance to bring more success to his boyhood team.
Carragher told the 'Stick to Football' podcast: "Where I’m at is that, as a local player and a Liverpool fan, Liverpool are in a position where they’re competing for the league and a European Cup every year. So, if you’re a Liverpool fan and a local player, I’m just disappointed that he doesn’t think that this is it. I’m putting myself in his position, where I’d be thinking that we could go level with Manchester United in league trophies, I’ve got another seven or eight years at Liverpool and want to get three or four ahead of them, with a couple more European Cups.
"We always talk about rivalries and never think of it as European. In some ways, Real Madrid is a rival, and no one will catch them – they are the biggest and most famous team - but I’m desperate for Liverpool to get second on that list – AC Milan are on 7 [European titles]. If I was him, I’d be thinking that we’ve got a great opportunity to win the league this year and could do this and that and at the end of his time, put Liverpool in a position [to top others]."
Alexander-Arnold will leave the Reds on a free transfer but Carragher has no issue with this as he came through the club's youth academy.
The former Liverpool defender said: "I’m not angry about him going on a free. In an ideal world, Liverpool would get £70m or £80m for him to reinvest, but you can’t have it both ways as a supporter. If Liverpool sign someone on a free in the summer, we’d all be delighted, so does that mean that his former club should absolutely hate him?
"Also, when you’re a local player you don’t cost anything – Trent going on a free is like Virgil van Dijk leaving for £75m, we’d still break even."