The Spanish club are looking to bring in a new right-back with Tottenham's Pedro Porro, Bayer Leverkusen's Jeremie Frimpong and Juanlu of Sevilla on their radar, according to The Athletic.
The report adds that sources with close knowledge of Madrid's transfer plans claim Alexander-Arnold is the No 1 option, but only if he allows his contract to expire.
The full-back's deal at Anfield runs up until June 2025, meaning he will be a free agent at the end of the season and free to talk to other clubs from January.
Liverpool have been negotiating with the 26-year-old over an extension but there is plenty of uncertainty surrounding the situation.
Real Madrid have reportedly made Liverpool star Trent Alexander-Arnold a priority target
Speaking about his future while on international duty for England, he told ITV: 'I've entered the prime of my career right now and I don't want to be the player who only won trophies when he was young.'
Madrid right-back Dani Carvajal was taken off on a stretcher during Real's win at home to Villarreal earlier this month.
The right-back is set for a lengthy spell on the sidelines after suffering an anterior cruciate ligament injury.
Los Blancos are also believed to be on the hunt for a new centre-back and could make a move in January.
Former Man City defender Aymeric Laporte is one player they have their eyes on, The Athletic add in their report.
The 30-year-old currently plays with Cristiano Ronaldo at Al-Nassr and has a contract which runs until 2026.
Other players being tracked are Palmeiras' 18-year-old Brazilian Vitor Reis, who has also attracted interest from Arsenal, and RB Leipzig's 21-year-old France international Castello Lukeba.