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That has only invited more speculation about Real Madrid, but the vice-captain has been warned he would be throwing it all away if he moved to the Spanish capital.
Held up as a fairy story, Alexander-Arnold has won it all after emerging from the club's academy system, and until recently looked destined to one day take over the captaincy from Virgil van Dijk.
Such are his achievements at Anfield to date, Alexander-Arnold would already be guaranteed legend status if he ended up as a one-club man, regardless of any future silverware.
But leaving for nothing in order to join Real Madrid as he enters his peak years would unavoidably taint what has gone before.
Alexander-Arnold would still have a place in the Anfield annals, but it would not be alongside his hero Steven Gerrard.
Another Scouse sports star, Paddy Pimblett has discussed Alexander-Arnold's position.
But there's no doubt that Alexander-Arnold could etch his name into foklore.
However, like plenty of other Liverpool fans, Pimblett expressed fears that Alexander-Arnold may have already had his head turned.
"He could also leave if he wants to," Pimblett conceded.
Maybe it's a simple case of the money talking.
Or maybe Pimblett and others are wrong, and Alexander-Arnold still hasn't made his decision