The Liverpool right-back was the subject of an offer to the Reds at the start of the transfer window.
However, Liverpool are not interested in selling him midseason, while Alexander-Arnold also wants to wait until the summer.
He can then join Real as a free agent, unless he signs a new Liverpool deal.
According to Revelo, Madrid believe that a deal to sign him in January is not worth pursuing.
They will instead park their interest in the summer and hope that Alexander-Arnold joins them.