While no formal offer was submitted, the Spanish giants made an informal inquiry about the availability of the 26-year-old, according to Sky Sports.
Alexander-Arnold's current contract expires at the end of the season, and media in Spain have suggested that he has communicated his desire to join Real Madrid to Liverpool.
Starting January 1st, Real Madrid will be permitted to enter into direct negotiations with Alexander-Arnold regarding a pre-contract agreement.
Alexander-Arnold, along with Mohamed Salah and Virgil van Dijk, are currently in contract negotiations with Liverpool, as all three players are set to become free agents in the summer.
Van Dijk recently stated there is "no deadline" regarding his contract situation as the clock ticks down to the end of his Liverpool deal.
The Netherlands international's current contract expires at the end of the season and from January 1st he will be able to negotiate with overseas clubs.
However, it is understood the 33-year-old's preference is to stay at Anfield.
"There is no deadline at all. We will see what the future brings. Is there an update? No," he told reporters earlier this month.
However, Salah admitted he has little to offer on new contract news.
Salah was asked for an update after Liverpool's win at West Ham on Sunday night.
"No, we are far away from that," he responded when asked for contract news. "I don't want to put anything in the media.
"The only thing on my mind is I want Liverpool to win the league and I want to be part of that.
"I will do my best for the team to win the trophy. There is a few other teams catching up with us and we need to stay focused and humble and go again."