The 26-year-old is yet to extend his contract beyond the end of the current campaign and is understood to be a priority target for the Champions League winners, who are without Dani Carvajal for the rest of the season due to a severe knee injury.
Carvajal, now 32, excelled both domestically and internationally in the 2023-24 season, but Real Madrid are beginning to plan for life without the long-serving right-back, who turns 33 in January.
Alexander-Arnold is thought to be just one of a few targets on Real Madrid's shortlist, which also includes Bayer Leverkusen's Jeremie Frimpong and Tottenham Hotspur's Pedro Porro, but Carlo Ancelotti has earmarked him as the number one pick.
Alexander-Arnold shares a close relationship with Real Madrid midfielder Jude Bellingham, although he offered Liverpool fans hope that he would be willing to sign a new deal if the club could match his silverware ambitions.
Could Liverpool entertain Alexander-Arnold offers in January?
The Reds have made an exceptional start to life under Arne Slot and currently lead the way at the top of the Premier League table with 18 points from their opening seven games, as well as beating AC Milan and Bologna in their opening two Champions League battles.
However, Spanish outlet Marca reports that the Reds have accepted that their boyhood star is not going to be signing a new deal, and Real Madrid are aware of Liverpool's acceptance on the matter.
As a result, Los Blancos may test Liverpool's resolve during the January transfer window, when Alexander-Arnold will be free to open pre-contract talks with foreign clubs, but the Reds will still demand a "significant" sum.
Whether the reigning La Liga and Champions League holders pursue a winter move will hinge on how Ancelotti's side cope without Carvajal for the next couple of months, as another loyal servant - Lucas Vazquez - prepares to deputise for the time being.
Nevertheless, having allegedly come to terms with the fact that Alexander-Arnold will not be penning a contract extension, Liverpool will have no choice but to entertain January bids lest they risk losing him for nothing in 2025.
Who could Liverpool replace Alexander-Arnold with?
Firstly, how do Liverpool even go about replacing a right-back with over 100 direct goal contributions in his senior career? Alexander-Arnold's defending may be the subject of ridicule at times, but what he brings to the Reds in the final third is invaluable.
During Jurgen Klopp's farewell season, the youthful Conor Bradley burst onto the scene during Alexander-Arnold's enforced absences, and the seldom-seen Joe Gomez is another option on that side of the field.
Gomez's own future is the subject of much uncertainty, though, and Bradley cannot be expected to hold the fort all by himself, so the powers-that-be should no doubt target a direct successor to Alexander-Arnold either in January or the summer.
Sevilla's Juanlu Sanchez - who has also been linked with Real Madrid - is thought to be on Liverpool's radar, as is Fiorentina's Michael Kayode, whom they could try to prise away from Serie A during the January window.
For now, Reds boss Slot will no doubt be forced to bat away more questions about the futures of Alexander-Arnold, Mohamed Salah and Virgil van Dijk before Sunday's mouthwatering Premier League clash with Chelsea.