Txiki Begiristain, City's director of football, hopes to conclude negotiations before the preseason begins, according to The Telegraph's James Ducker.
The fee would narrowly surpass the £60 million City spent on Riyad Mahrez last summer.
With Fernandinho turning 34 years old in May and Ilkay Gundogan nearing the end of his contract, Rodri has emerged as City's biggest summer target.
The 22-year-old is considered the natural heir to Spanish counterpart Sergio Busquets, possessing the technical ability, passing range, and composure to master the holding position.
"He can reach Busquets' level and maybe surpass it," Spanish national team coach Luis Enrique said in October. "Who knows what limits he has."
Rodri returned to Atletico for €20 million last summer after a formative spell at Villarreal. Making 28 starts in La Liga this season, he immediately assumed responsibilities in the heart of Atletico's midfield.
City had grown accustomed to failure in their long-winded pursuit of an influential pivot. Chelsea beat the reigning Premier League champions to Italian metronome Jorginho last summer, and Brazilian midfielder Fred preferred a move to crosstown rivals Manchester United.
The Citizens were reportedly in the running for Ajax sensation Frenkie de Jong before Barcelona swooped in. The Blaugrana agreed to sign the 21-year-old for €75 million in January.