Neither team gave the transfer fee, but Spanish media have spoken of a figure of between EUR75 million and EUR80 million. Rodriguez, 23, scored six goals for Colombia at the World Cup to win the Golden Boot award.
The 23-year-old will be unveiled at Real Madrid’s Santiago Bernabeu stadium at 8:00 p.m. (1800 GMT), the club said in a statement.
Shortly before the announcement, Rodriguez posed for pictures and gave the thumbs up sign as he left a Madrid clinic where he underwent medical tests.
“I am very happy. We will talk later,” he told reporters.
He had only been at Monaco for one year but his sizzling performance at the World Cup made him the number one target for many European clubs.
In a FIFA.com poll released Monday, his volley against Uruguay in the last 16 was declared goal of the tournament by more than four million voters — three quarters of those who took part.
The goal established the global reputation of the young playmaker, who cost Monaco 45 million euros (60 million dollars) when he signed from FC Porto of Portugal.