Los Blancos are not only ready to sign him on a free transfer in the summer, but put in a bid this month to secure the right-back’s services.
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Carragher, who hopes that Alexander-Arnold will sign a new deal at Anfield, feels the bid would not have come without the defender’s consent.
“They will have known what was happening,” said Carragher on Sky Sports, referring to Alexander-Arnold’s representatives.
“I’m not naive enough to think that Real Madrid would have put a bid in in January, for a player who they can get for nothing in four months.”
He added: “Do you seriously believe, when this bid comes in that they’re sat at home and see it on Sky Sports and say ‘wow!’’”