Rodriguez, who was linked with Manchester United before he joined Monaco last summer, impressed this summer at the 2014 World Cup in Br azil and has already admitted he is keen on playing in Spain at some point in the future.
A host of top European clubs are understood to have put the 22-year-old Colombia international at the top of their wish-lists, but Rodriguez says a move to the Bernabeu to join Los Blancos would fulfil a lifelong ambition.
"It's really flattering to know that a club of Real Madrid's stature is interested in me," he told Marca.
"It'd be a dream to join the club of my dreams. I have a great deal of respect for other clubs and admire the good football they play, but ultimately Real Madrid are Real Madrid."