Padrós was a founding member of Real Madrid and later served as club President between January 1904 and 1908, succeeding his brother Juan.

Padrós and his brother Juan were born into a Catalan family living in Madrid.

In 1902, Carlos became president of the Federación Madrileña de Foot-Ball and while in this post he suggested a football competition to celebrate the coronation of Alfonso XIII.

The game, which ended as a 1-1 draw, was arrange to celebrate the visit of French president, Émile Loubet.

During his presidency the club also won the Copa del Rey four times in a row between 1905 and 1908.