Los Blancos have identified Alexander-Arnold as one of their top targets as they look to bolster the right side of their defence in 2025.
The England international is out of contract at the end of the season and is yet to agree a new deal with his boyhood club.
There has been a suggestion that Real Madrid could try to pull off a cut-price January deal following Dani Carvajal's season-ending injury.
However, Real Madrid boss Carlo Ancelotti recently stated that the club have no plans to sign a new right-back in January, insisting that he has full faith in Lucas Vazquez and Eder Militao to cover the position.
Real Madrid to make Alexander-Arnold bumper contract offer
Even if Real Madrid decide against a January move, they are likely to make an effort to sign Alexander-Arnold on a free transfer at the end of the current season.
As it stands, the reigning European champions can start talks with Alexander-Arnold in January over a pre-contract agreement.
According to Caught Offside, Real Madrid are preparing to make the right-back a huge contract offer to make the switch to the Spanish capital.
The report claims that Los Blancos are willing to almost double the player's current salary by proposing a £14m to £15m a year deal.
The proposed contract offer would see Alexander-Arnold sign with the Spanish giants until the end of the 2029-30 season.
Why are Real Madrid keen to sign Alexander-Arnold?
Prior to Carvajal's serious knee injury, Real Madrid were believed to be looking for a new right-back to be his long-term successor.
A couple of shaky defensive performances from Vazquez has only heightened Real Madrid's need to strengthen the right side of their defence.
Real Madrid may have to remain patient until next summer, but it would certainly be worth the wait if they are able to sign someone of Alexander-Arnold's calibre on a free transfer.
If they were unable to pull off the move, Los Blancos would be getting a player who has established himself as one of Europe's top right-backs during his 321 appearances as a Liverpool player.
While he would not operate in the same position as Toni Kroos, Alexander-Arnold's impressive passing range would also help replace some of the quality lost by the German's departure earlier this year.