The club and the Germany international's representatives are still working on aspects of the deal, but Rudiger, 29, is now on track to sign for Madrid on a four-year contract, sources said.
ESPN reported last week that Madrid were looking at the possibility of signing Rudiger -- having distanced themselves from a deal earlier in the season -- after club executives were impressed by his performances in their Champions League quarterfinal tie with Chelsea.
Rudiger scored Chelsea's second goal at the Bernabeu on April 12 in a dramatic 3-2 win that saw the Premier League club eliminated 5-4 on aggregate after extra time.
Madrid began working on a agreement for the defender after that second leg, and sources said the deal could now be sealed "in record time."
Negotiations have moved fast, sources said, because Rudiger is keen to join Madrid, and has also considerably reduced his wage demands.
Earlier this season Madrid had ruled out the signing because they did not want to get involved in an expensive bidding war for Rudiger with other top clubs.
The situation changed in recent weeks, and Madrid decided to move after Carlo Ancelotti's team had problems forming a reliable back four in key games at this stage of the season.
Rudiger's arrival would see the team strengthen at centre-back with a top-level player, arriving without a transfer fee, just as they did with the signing of David Alaba from Bayern Munich last summer.
It would also allow Ancelotti to deploy Alaba at left-back if he wishes, pairing Rudiger with Eder Militao at centre-back.
ESPN reported on Saturday that Rudiger had rejected Chelsea's latest contract renewal offer -- worth over £200,000 a week -- and coach Thomas Tuchel confirmed on Sunday that the defender would be leaving Stamford Bridge.
"The situation is that he wants to leave the club," Tuchel said. "He informed me of this is in a private talk. We gave everything -- me and the club -- but we could not fight anymore.
"Without the sanctions we would at least be able to carry on fighting, but our hands are tied. We don't take it personally. It is his decision."