Manchester City have already brought in Elliot Anderson this summer and are working on a deal for Lille's Ayyoub Bouaddi, but the futures of some of their existing squad remain in flux.
Rodri, who has one year left on his contract at the Etihad, offered terms on a new deal back in April that he and his representatives are yet to accept, with a return to Madrid instead understood to be his preference.
The 30-year-old has been strengthened in that ambition by leading Spain to World Cup glory this summer, a triumph that has increased support internally at Real Madrid for a move to be sanctioned. Perez had initially been reluctant to back a deal, though others at the club are said to believe signing Rodri would benefit both the squad and Real Madrid's wider image.
Jose Mourinho, brought in partly to address a perceived divisiveness within the Real Madrid dressing room, has already pushed for a number of senior recruits since taking charge, and Rodri is seen as fitting the profile the head coach wants to build around.
Any deal for the Spain international, though, could have knock-on consequences for others already at the club, with further departures possible should Rodri's move go through.
Rodri's stock has rarely been higher after his performances for Spain over the summer, where he captained the side to World Cup glory and was named the tournament's outstanding player.
That form has only intensified the sense in Madrid that he could immediately slot into the void left in central midfield, though City are not expected to make his exit straightforward given his importance to the side and the fact he still has a year to run on his current terms.
Any deal, should it materialise, would represent a major statement of intent from Mourinho as he looks to reshape the Real Madrid squad in his own image following his return to the club.