The report describes the pursuit as "difficult" given Álvarez's profile, but Köln are understood to hold one significant advantage: West Ham United, who were relegated from the Premier League last season, are said to be fundamentally open to letting the 28-year-old leave immediately if the right approach comes in.
Álvarez, who made four appearances at this summer's World Cup on home soil for Mexico and started twice, is under contract with West Ham until 2028.
The versatile defensive midfielder moved to Europe with Ajax Amsterdam in 2019 before completing a switch to West Ham in 2023 for a fee in the region of €38m, going on to make 73 appearances for the Hammers before spending last season on loan in Turkey with Fenerbahce, managing one assist in a campaign disrupted by injury.
West Ham's relegation to the Championship has reshaped their squad planning, and Álvarez, now back at the club following the end of his loan, is among those whose futures look uncertain as the Hammers reassess their options at a lower level.
That situation, combined with Cologne's ambition to add proven quality to their midfield, could open the door to a deal that would have seemed unlikely while West Ham remained a Premier League club.
A 103-times capped international for Mexico, Álvarez has established himself as one of the most experienced holding midfielders outside Europe's traditional powerhouse leagues, making his profile an appealing one for a Köln side looking to add steel and know-how to their midfield ahead of the new Bundesliga season.
Whether the move can be squared financially remains to be seen, but the willingness on West Ham's side to sanction a departure gives Köln genuine hope of pulling off an ambitious piece of business this summer.