With just over two months remaining until the January transfer window, the right-back is still yet to extend his contract at Anfield past June 2025.
That is naturally leading to widespread speculation regarding his future with the option of signing fresh terms or taking on a new challenge each realistic possibilities.
Real Madrid's situation on the right-hand side of their defence has made the England international a natural target, particularly when he is in line to become available on a free transfer.
As well as Dani Carvajal being sidelined in the long term with a serious knee injury, both he and Lucas Vazquez are at least 32 years of age.
Furthermore, Los Blancos suffered a humiliating 4-0 defeat to rivals Barcelona in the El Clasico on Saturday evening, emphasising that upgrades may be required at the back.
Who could join the race?
Should Alexander-Arnold make the switch to the Bernabeu, there would need to be an acceptance that he would be in direct competition with a player in Carvajal who would likely be restored as first choice once he has fully recovered from a serious knee injury at the back end of 2025.
Nevertheless, according to Fichajes, the European champions could face competition from Paris Saint-Germain over the coming months.
The report alleges that the Ligue 1 giants are also in the running for Alexander-Arnold and discussions at some level have already taken place.
PSG's hierarchy are said to be convinced that the 26-year-old can bring extra quality to their defence and out wide as they bid to finally end their wait to win the Champions League.
Would Alexander-Arnold be first choice at Parc des Princes?
Achraf Hakimi is already a standout option for PSG at right-back, the Morocco star chipping in with two goals and three assists from his 10 appearances this season.
With 16 goals and 22 assists from his 130 outings in all competitions, the 25-year-old is cemented in the PSG defence under Luis Enrique, yet there will be an argument that Alexander-Arnold would produce similar numbers over that timeframe.
Meanwhile, Hakimi is in a position where he has less than two years remaining on his PSG contract, creating a scenario where they could plausibly cash in on his signature and acquire a like-for-like replacement on a free transfer.
That is something which may appeal to the club's owners if they have to find a way to create extra leeway with any of the financial regulations.