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The offer presented would've made Alexander-Arnold the best paid right-back in the Premier League and one of the highest-earning defenders in the world.
Despite negotiations continuing, Alexander-Arnold informed the club, where he's spent the last 20 years, that he wanted a new challenge.
European champions Real made an approach for the England international on New Year's Eve, but their interest was firmly rebuffed by Liverpool.
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Anfield chiefs held out hope that they'd be able to convince Alexander-Arnold to stay, with the Reds still in all four competitions at the beginning of January and the overwhelming favourites to win the Premier League title.
"After 20 years at Liverpool Football Club, I'll be leaving at the end of the season," he said in a video posted on social media.
"I think the decision for me is personal.
I'll love it forever and I'll forever be in debt to the club and the fans for everything you've done for me.
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"There's no doubt about it, I'll be back.
I'll be supporting, I'll be wanting the club to win, wanting the club to be successful.
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