Wirtz and Alonso previously worked together at Bayer Leverkusen, where they won the Bundesliga title last year.
Wirtz joined Liverpool this summer, while Alonso moved to Real Madrid.
As Wirtz faces challenges adapting to his new team, Alonso has asked Real Madrid's management to monitor Wirtz's progress.
Bayer sold Wirtz to Liverpool for £125m, but he has struggled to make a significant impact this season.
Alonso believes Wirtz would be a good fit for Real Madrid and would thrive under his coaching.
