The Brazil international's father visited the Etihad Campus towards the end of last season as Rodrygo was considering his options at Real.
City boss Pep Guardiola is a fan of the attacking midfielder, though their transfer plans shifted elsewhere ahead of the summer's Club World Cup.
Now City are renewing their interest in Rodrygo as Savinho pushes to join Tottenham this week.
The young winger wants to make the move to Spurs, where he is convinced he will enjoy better playing chances.
As such, if City sell Savinho, Guardiola will likely approve a move to bring Rodygo to England before the summer transfer market closes.