While the Bosman switch won't formally happen until his Blues deal expires on June 30, Real are expected to announce the signing in the coming days.
Italian transfer expert Fabrizio Romano said: "Real Madrid are set to announce Toni Rüdiger as first signing for the next season. Medical already completed in London weeks ago and contract signed.
"Rüdiger's agreement with Real will be valid until June 2026.
"No option for further season."