The Spanish giants are not obliged to register an interest in Alexander-Arnold as he can sign a pre-contract agreement with an overseas club in January.
However, sources close to the European champions say they want to be respectful to their Merseyside counterparts.
Liverpool are still hoping to convince England defender Alexander-Arnold to sign a new deal at Anfield.
His contract, along with captain Virgil van Dijk and Mohamed Salah, will end next summer.
It's prompted a swirl of rumours about new deals and potential transfers for all three.
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Alexander-Arnold has been long-linked with a switch to Real Madrid.
Los Blancos are currently operating without their usual right-back - six-time Champions League winner Dani Carvajal.
He suffered an ACL injury in October and this appears to have accelerated a succession plan for the 32-year-old.
Links to Alexander-Arnold have also grown with the England international enjoying a strong friendship with Real superstar Jude Bellingham.
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Their connection had previously led Liverpool fans to hope Bellingham would join the Reds from former club Borussia Dortmund.
The 21-year-old, though, moved to the Spanish capital where he has already won LaLiga and the Champions League.
Losing Alexander-Arnold next summer might not be the loss Liverpool would have feared a year ago.
Conor Bradley stepped up in his absence against Real Madrid in the Champions League and preceded to nullify Kylian Mbappe in stunning style at Anfield.
One tackle in particular stunned Merseyside and sent shockwaves around Europe as he steamed in full throttle and snuffed out a dangerous run from the French World Cup winner.
Legendary England full-back Stuart Pearce, working for talkSPORT at Anfield, joked: "With Trent Alexander-Arnold sitting on the bench at the moment, he must be thinking to himself, I might just knock on the manager's door and say, please, have you got a contract for me?
"Because you've got someone who's been absolutely outstanding tonight, he's been brilliant.
"He's dealt with [Kylian] Mbappe, he's got forward, he's assisted goals, he's been everything."