The Bayer Leverkusen chief, 41, is tipped to succeed Carlo Ancelotti after the Italian’s contract runs out next summer.
Former Spain, Real and Bayern Munich midfielder Alonso would be a surprisingly inexperienced choice.
After retiring as a player in 2017, he managed Madrid’s youths before three years in charge of the Real Sociedad B team.
Admittedly, though, he’s been a huge success since landing the main position at Leverkusen 11 months ago.
After guiding them to sixth in the Bundesliga last season, Leverkusen sit second this term – behind champions Bayern only on goal difference.
Now Spain’s Radio MARCA say Madrid have identified Alonso for arguably the biggest job in club football.
And it’s claimed his appointment is all but inevitable.
Alonso is best-known in England for for his stylish stint at Anfield from 2004-9, winning the Champions League and FA Cup.
Outgoing Ancelotti has won the Champions League four times as a manager – twice with AC Milan, then once in each of his two spells at the Bernabeu.
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But his exit at the end of the current campaign has long been predicted.
It’s thought he will then take his first international job with Brazil, starting with the 2024 Copa America.