The 23-year-old Colombian underwent a medical with the European champions earlier today before joining the club for a fee believed to be in the region of £60m.
He is due to be presented to the Madrid fans at the Santiago Bernabeu this evening.
Rodriguez had been linked with a number of clubs across Europe following a string of impressive performances for Colombia at the World Cup.
The youngster finished the tournament as the Golden Boot winner with six goals to his name, though Colombia failed to get past the quarter-finals.
Rodriguez leaves Monaco following just one season with the Ligue 1 club.
