The Spanish club has allocated €100-150 million to bring in a top-class player before the new season begins.
Real Madrid is looking for a replacement for Toni Kroos, who retired last summer, and views Rodri as their primary target to address this issue.
Rodri is currently tied to Manchester City until 2027 and is recovering from knee surgery he underwent last year.
While other players are being considered, Rodri remains the top priority for Real Madrid.
Other names being discussed include Nicolò Barella from Inter Milan, Enzo Fernández from Chelsea, and Alexis Mac Allister from Liverpool.
However, all of these players have long-term contracts with their respective teams, making their departure challenging.