Rudiger is in the final year of his Chelsea contract and negotiations over a new deal have not gone according to plan.
The defender rejected an offer of €8.5m per season from Chelsea, but the Blues are yet to improve on those terms, which has angered the defender.
Bayern Munich are interested in signing him and the German is open to the prospect of joining the German champions next summer.
But according to Spanish daily Marca, Real Madrid are also keeping a close eye on the defender’s situation at Stamford Bridge.
The Spanish giants took up the opportunity to sign David Alaba on a free transfer last summer.
They are alive to the prospect of Rudiger being available as a free agent at the end of next season.
Real Madrid will keep a close eye on their wage structure and the needs of the squad before taking a call on the German.
Chelsea want to keep Rudiger but it remains to be seen whether they are prepared to fulfil his desire to become one of the highest-paid players in the squad.
