Los Blancos' most likely incomings this summer are Kylian Mbappé, Endrick and Alphonso Davies, which would leave little room for manoeuvre to make any other notable signings.
Even with Toni Kroos' retirement, they are unlikely to move for a midfielder with Jude Bellingham expected to drop deeper.
They are still looking at options for the future and Wirtz is most heavily favoured, with the his versatility adding to his appeal.
He has reached those heights under Alonso, somebody who has been seriously considered as the replacement for head coach Carlo Ancelotti, whenever the Italian leaves. Alonso also played for Real Madrid for five seasons.
However, Los Blancos are not in any rush to complete these moves despite knowing there is plenty of competition from across Europe. Wirtz has a contract that runs until 2027, but they will be watching every Leverkusen match to keep an eye on both Wirtz and Alonso.