After Los Blancos failed in their pursuit of Paris Saint-Germain forward Kylian Mbappe, attention turned to their next target, and the Spanish champions are believed to have overtaken Liverpool in the race to sign Tchouameni.
The 22-year-old continued to be a key figure in the Monaco side during his second campaign with the club, featuring 50 times across all competitions, netting five goals and registering three assists along the way this year.
According to The Mirror, Tchouameni has promised the Reds and Real Madrid that he will make his final decision in the coming days, but Jurgen Klopp's side fear they are second choice for the defensive midfielder.
With another two years on his contract at the Stade Louis II, whichever club is successful in persuading the young French international will have to pay a reported £67m to Monaco for his services.
In total, Tchouameni has scored eight goals and provided seven assists in 95 appearances for Monaco.