Real Madrid sign Golden Boot Rodriguez
Colombia's midfielder James Rodriguez runs with the ball during the 2014 FIFA World Cup quarter-final match between Brazil and Colombia at the Castelao Stadium in Fortaleza on July 4, 2014 - by Fabrice Coffrini
Neither team gave the transfer fee, but Spanish media reported a figure of between €75 million and €80 million ($101m-$108m), making him one of the most expensive players in history.
Rodriguez, 23, scored six goals for Colombia at the World Cup to win the Golden Boot award.
He 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 passed a medical test.
"I am very happy. We will talk later," he told reporters.
The Colombian striker is Real's second major signing in five days as they prepare for a new season of confrontation against eternal rivals Barcelona.
German midfielder Toni Kroos moved from Bayern Munich on Thursday, just days after the 24-year-old helped Germany win the World Cup.
Real have also been linked with a move for Keylor Navas, the goalkeeper whose heroics helped Costa Rica reach the World Cup quarter finals. Navas plays with Levante in the Spanish first division.
Rodriguez 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 World Cup by more than four million voters -- three quarters of those who took part.
Monaco paid €45 million ($60m) when he signed from FC Porto of Portugal.
Spanish sports daily Marca, which has close ties with Real Madrid's board, said Real will pay €75 million straight away and up to five million more in add ons.
If confirmed, the fee would make Rodriguez the fourth most expensive player in history behind Gareth Bale, Cristiano Ronaldo (both of Real) and Luis Suarez who joined Barcelona this month.
Rodriguez went to the World Cup as Colombia's second best known striker. But Radamel Falcao was injured in January leaving the starring role to his Monaco teammate as Colombia reached the last 16 for the first time.
Uruguay's coach Oscar Tabarez compared Rodriguez to Argentine legends Diego Maradona and Lionel Messi, the "players who succeed in doing things because they are special."
The Colombian has made it clear for a long time that it was the Spanish league that he wanted to get to.
"The Spanish league is a league that I've always watched -- it's different from the English League, which is very physical," he told FIFA.com in an interview in November.
"The game in Spain is more about technique, and about getting more touches on the ball. That's why I like it so much."
Rodriguez started his career at 16 for Colombian second division side Envigado FC and built up an immediate reputation.
In 2008, he was signed by Argetinian side Atletico Banfield where he became known as "James Bond of Banfield" after scoring an extraordinary left-footed lob for the unfashionable side in February 2010.
He moved to FC Porto in Portugal where more spectacular goals brought three league titles in three seasons. Monaco bought him in May 2013.
Rodriguez prides himself most in having been dubbed 'El Pibe Nueva' (the New Kid) in place of Carlos Valderrama, Colombia's star of the 1990s, who was called 'El Pibe'.
"It's an honour to be considered the heir to 'El Pibe' at my age," he told FIFA.com.
"Since I was little, I admired Valderrama and I always wanted to be like him. He was my hero and I am proud to wear the number 10 for Colombia while helping the national team return to the World Cup."