The pair were also spotted holidaying in Los Angeles together last month, where Alexander-Arnold was seen signing a Madrid shirt for a fan.
Anfield bosses also have to sort out what to do with Mo Salah, 32, and Virgil van Dijk , 33, who are the only two players at Anfield who earn more than Alexander-Arnold.
All three players contracts will expire in June 2025.
It still seems unlikely Liverpool would cash in on their academy graduate this summer as it would be a wrench for both parties.
But Real and Barca, along with other top clubs, are looking at what might be possible in the next 12 months or so.
Jude Bellingham and Trent Alexander Arnold holidayed together in LA Instragram
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His assist tally and chance creation from right back have led to calls for him to be utilised further forward despite misgivings about that from former boss Jurgen Klopp.
Alexander-Arnold is arguably the most complete passer in the game bar Kevin De Bruyne , and has scored 15 goals for Liverpool and provided 58 assists, a record for defenders in the Premier League.
New boss Arne Slot prefers a more orthodox right back, with Northern Ireland international Bradley mooted as the more natural fit there.
Meanwhile, Liverpool legend Michael Owen said Madrid might prove too irresistible of a destination for Alexander-Arnold.
Owen told Casino Hawks: “Obviously Real Madrid at the moment are the best team in Europe, they are the biggest team in the world and have that iconic kit and their players… It’s the best stage in Europe.
“I had a similar thing at that stage of my career, and I decided that it was an opportunity that I couldn’t pass up.
“I don’t know if it’s an opportunity for Trent or a possibility, but he’d have a lot to think about if they came in for him.”
Liverpool are the only Premier League club yet to sign a player in the summer transfer window but are expected to bring in a new defensive midfielder.
Slot snubbed Spurs, hates defending and has a PASS named after him
By Dan King
LIVERPOOL’S new manager Arne Slot was a good enough player to have a type of pass named after him.
But it is as a manager that the Feyenoord head coach is really making his mark.
Like another bald Dutchman, Manchester United boss Erik ten Hag, Slot earned his stripes bossing smaller sides before being given a chance at one of the Netherlands’ big three.
But the question Liverpool are pondering is whether Slot would make a better fist of running one of the biggest clubs in the world than Ten Hag so far has at Old Trafford.
Slot, 45, was certainly easier on the eye as a player than centre back Ten Hag.
“The Arne Slot Ball” was something he perfected as a silky No 10 – a back-to-goal, first-time, 180-degree spin and flick behind the defensive line for a winger to run on to deep in the opposition half.
Click here to read all about the incoming Liverpool boss.
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