According to Sergio Santos Chozas of AS, Rodri is "very near to agreeing to go to the Etihad Stadium" and has an offer on the table for either five or six years that would see his salary significantly increase.
Chozas added other clubs are interested in Rodri and the move to City is not yet a done deal, but the 22-year-old is attracted by the chance to work under Pep Guardiola.
Rodri has only been at Atleti for a year after moving to the Wanda Metropolitano Stadium from Villarreal last summer on a five-year deal.
The Spaniard has been impressive under Diego Simeone, playing in 33 of Atleti's 37 La Liga games this season in the central-midfield role.
He would be a fine long-term replacement for Fernandinho in the deep-lying midfield role where City lack depth:
Fernandinho has been one of City's most important players ever since he joined the club in 2013.
But he is now 34, and City need to find an adequate replacement or risk losing some control in the middle of the park.
City have been brilliantly successful in the transfer market recently, with the majority of players Guardiola has spent big money on settling well and thriving at the Etihad.
Rodri looks as though he would be another to add to the list of successful transfers should City manage to snap him up.
At £61 million, he would be the club's most expensive signing, eclipsing the £60 million they spent to take Riyad Mahrez from Leicester City.
But given that he boasts crucial attributes in defence and distribution, he looks an ideal successor for Fernandinho at only 22 years of age.