The 28-year-old has less than seven months remaining on his current contract at Stamford Bridge, and talks to extend his deal with the Blues have failed to progress.
According to the Independent, Rudiger's brother and agent Sahr Senesie held talks with representatives of Real Madrid over the last few weeks and an 'informal agreement' for the German to move to La Liga next summer has been reached.
The report adds that Premier League duo Manchester United and Tottenham Hotspur have also expressed their interest in signing Rudiger; however, the centre-back would prefer a move to the Santiago Bernabeu.
Rudiger, who is believed to be asking for an annual salary of €12m (£10.2m) to remain at Chelsea, has made 20 appearances across all competitions for Thomas Tuchel's side this season, registering two goals and two assists. body check tags ::