Los Blancos have been heavily linked with a move for Trent Alexander-Arnold in recent months, with the England international's contract at Anfield set to expire at the end of the season.
Van Dijk, like Alexander-Arnold, is also out of contract at Liverpool at the season's end and will be eligible to sign a pre-contract agreement with clubs outside of England from January 1.
Although the Dutchman has expressed his desire to stay with the Reds, it seems the La Liga champions are still keen on trying to persuade him to move to the Bernabeu.
Real Madrid consider Van Dijk ideal defensive addition
Los Merengues have been plagued by numerous defensive injuries this season, with the likes of Eder Militao, David Alaba and Dani Carvajal all sidelined for the foreseeable future with serious knee problems.
According to Football Transfers, due to concerns about the injury records of Militao and Alaba, Real Madrid are reportedly considering moving them on and Van Dijk is seen as an ideal addition to help strengthen their backline.
The 33-year-old turned down Liverpool's initial contract renewal, though that is widely believed to have been a negotiation strategy, and discussions are still ongoing to secure a new deal that would keep him at the club for the next season and beyond.
It is understood that the Dutchman and the club are keen to reach an agreement for a renewal, which is no surprise considering Van Dijk's impressive performances this season and his evident enjoyment playing for the club.
Van Dijk signed for Liverpool from Southampton in January 2018 for £75m, a then-world-record transfer fee for a defender.
Liverpool join race to sign Bayern star in the summer
While contract discussions around existing players are expected to continue over the coming weeks at Anfield, reports suggest that the Reds are also considering moves for the upcoming transfer window.
CaughtOffside revealed this week that Arne Slot's side are preparing to join the race to sign Bayern Munich left-back Alphonso Davies.
The Canadian international was widely expected to join Real Madrid on a free transfer at the end of the season, while Manchester United have also been rumoured to be interested in the 24-year-old's services.
The former Vancouver Whitecaps star's current terms are set to expire with the Bavarians in the summer of 2025, and his agent Nick Huoseh recently revealed that he will talk with any interested top club in the New Year.
Liverpool currently lead the Premier League standings by two points with a game in hand on second-placed Chelsea and will be back in action on Sunday when they travel to the capital to face Tottenham Hotspur.
Written by
Devesh Jaganath