The Reds have made it clear they are not willing to entertain losing Alexander-Arnold mid-season.
It's believed no figures were discussed but the move was still immediately rebuffed.
The England right-back is out of contract at the end of the season.
However, he can sign a pre-contract agreement with Real if he desires given they are a foreign club.
Liverpool are still hoping to persuade Alexander-Arnold to pen a new deal at Anfield.
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The 26-year-old is one of three top Reds stars all set to be out of contract in the summer.
Virgil van Dijk and Mohamed Salah are also set to see their deal expire in 2025.
However, the club are intent on getting new contracts sorted and extending their stays at Anfield.
The latest news comes as Liverpool end 2024 at the top of the Premier League table by eight points.
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They sit clear of unexpected high-flyers Nottingham Forest and nine ahead of traditional challengers Arsenal.
The club have been revitalised by new boss Arne Slot who joined the club from Feyenoord in the summer, replacing long-serving Jurgen Klopp.
With Klopp, Alexander-Arnold was a key component in a team that won the Champions League in 2019 and a Premier League title in 2020.
He's also won an FA Cup and League Cup with the Merseysiders.
The defender came through the Liverpool academy and made his debut in 2016, after being tipped for success by club legend Steven Gerrard.
He's since made 243 Premier League appearances for the club, making 62 assists and scoring 16 goals.
The 26-year-old is renowned for his technical ability, adept at crossing the ball from the right flank, and at set-pieces.
Alexander-Arnold is long-term target for Los Blancos who currently have a need for a new right-back.
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Club legend Dani Carvajal suffered a torn ACL in October with versatile right-sided star Lucas Vazquez his short-term replacement.
But even if 32-year-old Carvajal returns next season, there is a chance he will leave the Spanish giants.
Alexander-Arnold would likely receive a warm welcome in Madrid from fellow England international Jude Bellingham.
The pair appear to have struck up a firm friendship, despite Bellingham being five years his junior.
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Before the 21-year-old's move to the Bernabeu, it had been thought he could even find himself on Merseyside with his pal.
However, Liverpool now face a real fight to retain his services.