The Ligue 1 club does not want to sell the leading scorer of Brazil 2014, although it has taken heed of Rodriguez's desire to join Madrid, with the 23-year-old having recently stated that he would go to the Spanish capital with his eyes closed.
As such, Monaco has floated a price of 70 million pounds to the Champions League winner, a sum that would place the Colombia international just behind prospective teammates Gareth Bale and Cristiano Ronaldo in the list of world-record transfer fees.
Goal understands that Manchester United could compete with the big-spending Liga club for Rodriguez's signature and, while Madrid president Florentino Perez will attempt to negotiate the price down, the numbers are considered viable for a Galactico signing within the Santiago Bernabeu.
Germany midfielder Toni Kroos, meanwhile, has secured a move from Bayern Munich to Real Madrid.
As a result, Sami Khedira and Angel di Maria have both been placed on the market despite their contributions to Madrid's Champions League triumph, with Arsenal and United monitoring their situations.
But it is Rodriguez who could be Madrid's major Galactico signing of the summer, with club president Florentino Perez buoyed by the success of last summer's record acquisition of Bale.
The former Porto star scored 10 times in 38 appearances for Monaco last season as he helped the club qualify for the Champions League and lit up the World Cup with his performances for Colombia, which was eventually knocked out by Brazil at the quarterfinal stage.