The 29-year-old is out of contract this summer and has rejected the offer of a new deal, which is understood to have reached in excess of £200,000-a-week.
Chelsea's ability to negotiate has been hampered by restrictions imposed following the U.K. government imposing sanctions on outgoing owner Roman Abramovich over his ties with Russia president Vladimir Putin.
Under a special licence granted to the club to continue operating while a sale is negotiated, Chelsea cannot agree any fresh contracts with any players which created a window of opportunity that Real and other clubs have looked to exploit.
Sources have told ESPN that Rudiger has offers on the table in excess of the terms Chelsea were offering and an extension was complicated further by agent demands over a high fee for his representatives and a signing-on fee in the region of £10 million.
ESPN reported on April 18 that senior figures at Real were impressed by Rudiger's performances over the two legs of their Champions League quarterfinal tie against Chelsea and talks are thought to have accelerated in recent days.
An agreement with Real is yet to be finalised -- Manchester United are among a number of other clubs to have been linked with Rudiger -- but Chelsea are now resigned to losing the centre-back at the end of the season.
Andreas Christensen is also expected to depart with Barcelona in advanced talks to sign the defender while Chelsea triggered a one-year extension to tie Cesar Azpilicueta to the club but he could yet move to the Catalan giants.