The Principality outfit had to pay an estimated €6.6 million ($6.9 million) for the Rodriguez transfer, according to French news site Mediapart, one of the 12 European media organisations aiming to expose financial fraud in the game through Football Leaks.
Monaco signed Rodriguez for €45 million from Porto in 2013 and sold him for a reported 80 million euros to Real a year later.
The club also paid the Spanish taxman 5.1 million euros in tax on profits from the sales of Belgian Yannick Carrasco to Atletico Madrid and Tunisian Aymen Abdennour to Valencia.
Spain views the Principality of Monaco, where the Ligue 1 club is based, as a tax haven and imposes a 19 percent tax on profits from sales into the Spanish market.
Contacted by Mediapart, Monaco confirmed that a "litigation is in process" but did not comment on the sums reported.
"The amounts were paid but are contested," said Monaco's management.
Mediapart put the total amount Monaco have paid Spain's tax authorities at around €11.7 million.
The Football Leaks investigation has also accused Real Madrid star Cristiano Ronaldo and Manchester United manager Jose Mourinho of tax evasion - charges the pair deny.
The wide-ranging probe of financial documents accused Manchester United and France midfielder Paul Pogba of stashing away money in the Channel Islands, at the request of his agent Mino Raiola, and claimed Paris Saint-Germain duo Angel Di Maria and Javier Pastore funnelled parts of their income through tax havens.
On Tuesday, France's national financial prosecutor's office said they had opened a preliminary inquiry into tax fraud and money laundering in the wake of the Football Leaks revelations.