STORY: Reuters TV footage showed riot police clashing after people threw glass bottles in Paseo de Recoletos, leaving at least one person injured.
Late on Saturday, thousands gathered at Cibeles square to celebrate the title with no incidents.
Los Blancos secured a record-extending 14th European Cup when they beat Liverpool 1-0 in Paris.
The game started after a 36-minute delay following incidents outside the stadium when fans were tear-gassed by riot police as people without tickets tried to force their way in.
Liverpool had won the previous European Cup final both teams had played in Paris, winning 1-0 at the Parc des Princes in 1981, which was Real's last defeat in a final.