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The 26-year-old had been at the club for two decades, coming through the academy and playing 352 times for the club - who he helped to win nine major honours, including two Premier League titles and the 2019 Champions League.
"He will lose something that he's got right now in terms of Liverpool supporters' love for him, him being 'one of their own' will diminish," said Carragher to Sky Sports.
"Some people will say: playing for Real Madrid, that price is not worth paying.
Some other people will say: 'You've got to play for Real Madrid, it's the biggest club in the world.'
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I don't think any Liverpool fan can deny that Real Madrid are the biggest club in the world.
"But that [love from the fans] is something he has sacrificed playing for Real Madrid."
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Asked if that love and devotion from the fans is an important thing for a player, Carragher replied: "It mattered to me, but I was never in this situation where I had the options of maybe choosing a club of the calibre of Real Madrid.
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Some people want, in their career, to play for some of the best clubs in the world - and you look at that career and think: 'Wow - Liverpool and Real Madrid'.
"I'm actually thinking about Xabi Alonso's career - Liverpool, Real Madrid and Bayern Munich - some of the biggest clubs in the world.
He came through the academy, he cost the club nothing.
"Liverpool were never going to sell Trent Alexander-Arnold before his contract was up and Real Madrid were never going to sign Alexander-Arnold before his contract was up.
"Because if you look at the history of Real Madrid signings over the last two or three years, I think they speak to players and their agents a year or two out - and say if you want to play for Real Madrid, we will sign you in a couple of years.
"I'm not saying that's what happened with Trent, but it definitely happened with players before and may have happened in this situation.
"But Real Madrid don't pay massive fees, they wait for players to leave on a free
And then they buy someone who fills in around him, whether that may be a utility player, maybe a centre-back who can play right-back?
"Or do they go and buy a recognised right-back and it's a real fight for the right-back position?
"I think Bradley - for what he's done so far in a Liverpool shirt - deserves to be given a go at being our right-back