The Reds defender is out of contract in the summer and can speak to clubs overseas as of January ahead of a potential free transfer.
Alexander-Arnold was linked with a switch to Madrid earlier this year and, with no new deal agreed at Liverpool, speculation has returned.
Reports from Spain claim that the England international has told his current club that he wants to leave for Madrid.
However, talkSPORT understands that these are premature, with negotiations over a new deal at Anfield still ongoing.
Alexander-Arnold recently insisted that details over a potential new contract at his boyhood club will be kept under wraps.
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A move to Madrid would see him become the latest Galactico signing after Kylian Mbappe and Jude Bellingham.
However, it would bring an end to a 20-year spell at Liverpool during which he has so far made 332 appearances across all competitions.
Alexander-Arnold was pivotal in their Champions League and Premier League successes in 2019 and 2020 respectively.
Most of his Liverpool career has come with Jurgen Klopp at the helm, with the right-back tearful after the German's final match in May.
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He could be one of three major stars to leave at the end of the season as Mohamed Salah and Virgil van Dijk also see their deals end in June.
Uncertainty over their futures continue to linger, with Arne Slot not willing to provide any updates on their situations.
"In general, I don't talk about contract situations here," he said during the week.
"It would be a surprise if I was the one to announce that one of them has extended, so I think that's in a different moment.
"But just, yeah, keep asking because that's your job, but you're probably not going to get an answer from me.
"But, it's clear that from January 1, maybe for you guys things change a lot, but they are in constant talks with the club and lets wait and see what happens.
"As long as they keep performing like this, the manager or the head coach, I have to say, is happy with them."