City believe the 22-year-old is the ideal candidate to strengthen their midfield and are prepared to pay his buy-out clause, which is thought to be €70 million (£60m/$80m).
City sporting director Txiki Begiristain was in attendance at Atletico’s Wanda Metropolitano stadium on Tuesday night for the Madrid club’s clash with Girona, who belong to the City Football Group.
Other leading European clubs are interested in Rodri, but City are especially determined to secure his signature having missed out on several high-profile targets in the last 18 months.
City boss Pep Guardiola is still frustrated by his club’s failed attempt to sign Frenkie de Jong from Ajax, which followed on the heels of Jorginho and Fred electing to go to Chelsea and Manchester United, respectively.
Begiristain, who is in charge of City’s transfer negotiations, is acutely aware that Rodri is now the preferred option of Guardiola and his coaching staff.
That means that interest in Lyon’s Tanguy Ndombele, who City had started scouting seriously over a year ago, has cooled, and there is a suspicion at the Etihad Stadium that the France international may have his heart set on a move elsewhere in any case.
City sources have also ruled out West Ham’s Declan Rice as a target this summer.
Goal understands that City had been keeping tabs on Chelsea’s N’Golo Kante and had asked intermediaries to keep them abreast of his situation at Stamford Bridge, in the event that he became unhappy with life under Maurizio Sarri.
A concrete approach was always regarded as unlikely, and Chelsea’s transfer ban, which they have appealed against, ensured City finally gave up any lingering hope of signing the Frenchman.
A transfer ban could also scupper City’s summer plans, although it is understood that FIFA is yet to complete its investigation into alleged breaches of the rules governing the signing of overseas youth players.
Should City be able to sign players this summer, they will also make moves to bring in a new forward. Goal revealed in February that the Blues have been closely monitoring Frankfurt's Luka Jovic, although they face rival interest from Barcelona, among other clubs.
Benfica's Joao Felix is also of interest, although his release clause of €120m is thought to be prohibitively high.