The England international has already claimed three assists from six appearances in all competitions this season and is not due to become a free agent until 2025.
However, Real supposedly recognise that a swoop for Alexander-Arnold in the summer of 2022 will not be simple given his recent extension, and Liverpool would not consider offers lower than £64m for the 23-year-old.
The Premier League and Champions League-winning right-back has already made 185 appearances for the Reds across all competitions, scoring 10 goals and setting up 47 more.
Alexander-Arnold has been struggling to overcome a groin problem in recent weeks but is expected to be fit for Saturday's Premier League encounter with Watford. body check tags ::