The Colombia star is one of the game's hottest properties after a superb showing at the FIFA World Cup in Brazil, where he remains top goalscorer with just two games to be played.
Rodriguez's six goals included an astonishing chest-and-volley effort against Uruguay in the second round and he is a candidate for the Golden Ball award after scoring in all five of his games.
Rodriguez scored 10 and assisted another 14 goals with Monaco last term after a €45 million move from Porto, but has opened the door for Real to make a bid after expressing his desire to turn out at the Santiago Bernabeu
"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.
"I'm eternally grateful to [Monaco president] Dimitry [Rybolovlev] and [vice-president] Vadim Vasilyev for the opportunity they gave me. The club is building something big.
"Obviously it would be a dream for me to play for Real Madrid, but I will always respect my current club, who will have the last word on the matter.
"Would I jump at the chance to go to Real? No doubt about it, my answer is a resounding yes."
Talk of Rodriguez moving to Real has intensified after his stunning displays for Colombia in Brazil.
Though the playmaker was happy with his individual performances, he believes more could have been done to keep his country in the competition.
"It's true that I played well at the World Cup, but I can't help feeling regret that we didn't make it to the final.
"It's an opportunity that may not come around again too easily. We dreamed of making it further, which is why I cried the day we were knocked out - such was my sadness at exiting the tournament. But that's football."
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