Alexander-Arnold's Liverpool contract expires at the end of June and he is yet to agree to fresh terms.
The Reds have three players due to be out of contract, with Mohamed Salah and Virgil van Dijk being the other two stars.
The 26-year-old has yet to publicly comment on the situation, despite reports from The Times that Liverpool rejected a bid from Real believed to be worth £20 million to sign him in January.
Liverpool's stance appears to be that he is not for sale as they pursue their first Premier League title in five years under new manager Arne Slot.
Real, though, appear to be cracking on with other business in the meantime as the Alexander-Arnold saga rumbles on.
According to Marca, Los Blancos have agreed personal terms with Bayern Munich left-back Alphonso Davies.
The Canada star's contract expires this summer, with his situation parallel to that of Alexander-Arnold's in that his current club have been in talks for him to stay.
It is claimed that Real Madrid are 'closer' to receiving the final green light from Davies and his representatives, with the player's agent Nedal Househ reportedly 'giving the order' for lawyers to draw up contracts based on the agreed terms.
Carlo Ancelotti's side have various difficulties in the left-back position, with Fran Garcia reportedly only seen as a rotation option and Ferland Mendy experienced a mixed run of form and several injuries over recent years.
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