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They may expect him to remain in Germany with current club Bayer Leverkusen or transfer rivals Bayern Munich, but until the player's decision is made Anfield chiefs are going to do all they possibly can.
The pool of suitors shrank when Manchester City officially exited the race, though reports from Spain have suggested that LaLiga giants Real are still on the periphery.
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If the expected new Los Blancos manager, former Red Xabi Alonso, has indeed asked Florentino Perez to sign Wirtz then the 'communicado oficial' serves as a bad piece of news.
Midfielder Luka Modric, 39, has penned a lengthy, emotional statement to confirm that his career in the Spanish capital is to come to an end after 13 years.
Over that time he has become the club's most highly decorated individual and has been their chief creative spark, scoring 43 goal and assisting 95 times.
So demand for a player of a similar ilk - not only playing style but also with their career ahead of them with time enough to become the next club legend - is set to increase as Alonso sweeps in a new era at the Santiago Bernabeu.
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This is in line with Arne Slot's aim to take a firm, controlling hold of football in England with a bolstered Liverpool squad.
A report in Spain claims that Alonso has already spoken highly about Wirtz to his future bosses in Madrid.
Defensa Central reported that in a letter of introduction, Alonso said about Wirtz: "He is the best in his position, he gives us things we do not have.
He can win the Ballon d’Or."
For the past two-and-a-half seasons at Leverkusen, under Alonso, the German international has made 119 appearances, scoring 38 goals and assisting 43 times.
It is evident that the pair have a strong bond as player and coach, with Wirtz writing a message to the Spaniard upon his departure from the Bundesliga: "Thank you for everything coach!
Wish you all the best for your next chapter!"
In return, Alonso told the German media of the 22-year-old: "Flo is one of the top players in the world.
It's normal in football that there are rumours, especially with such a special player.
"There were rumours (in the past), there are rumours today and there will be rumours in the future.
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