Alexander-Arnold is set to be out of contract at the end of the season, and the LaLiga giants have been lurking to snap up the right-back.
According to The Athletic, Real are 'working closely' to complete a pre-contract agreement and must inform Liverpool if a deal is struck.
Los Blancos also had a formal approach to sign Alexander-Arnold, 26, turned away in January.
Since his breakthrough to Liverpool's first team from the academy in 2016, Alexander-Arnold has emerged as one of the most regarded right-backs in the world.
In 349 appearances so far, he has scored 22 goals and provided 87 assists as he flourished under the guidance of former manager Jurgen Klopp.
Alexander-Arnold played a role in helping Liverpool win the Premier League title in 2019/20 - the season where he was also named the PFA Young Player of the Year.
He has also won the Champions League, FA Cup each on one occasion and two League Cups.
Alexander-Arnold is among three key Liverpool players whose contracts expire in the summer, alongside top scorer Mohamed Salah and captain Virgil van Dijk.