The 29-year-old joined the Blues from Roma in the summer of 2017 on a five-year deal, and those terms now have less than two months left to run.
Chelsea made multiple efforts to try to tie Rudiger down to fresh terms this year, with the Germany international often one of the first names on Thomas Tuchel's team sheet.
However, the Blues' efforts to negotiate a new deal with the centre-back did not bear fruit, and Tuchel recently confirmed that he would leave when his deal expired in the summer.
"The situation is that he wants to leave the club. He informed me of this is in a private talk. We gave everything - me and the club - but we could not fight anymore because of the actions. Without the sanctions we would at least be able to carry on fighting, but out hands are tied. We don't take it personally. It is his decision," Tuchel told Sky Sports News.
Despite links with Manchester United, Barcelona and Juventus, Real Madrid have long been regarded as the front-runners to land Rudiger as a free agent in the summer transfer window.
However, transfer expert Fabrizio Romano now claims that Rudiger is poised to sign a four-year contract with the capital giants, who will officially announce his signing at the end of the season.
"The next good news for Real Madrid will be Toni Rudiger's signature on the official contract until June 2026, matter of days," Romano wrote on Twitter.
"Club statement expected at the end of the season but the deal is already done and sealed. Rudiger will play for Real Madrid."
Even though Rudiger will be arriving at the Bernabeu as a free agent, the deal will supposedly still cost Real in excess of €59m (£50m), with the defender pocketing €11m (£9.4m) in yearly wages.
Despite the uncertainty surrounding his future, Tuchel has placed his faith in Rudiger throughout the season, and the former Stuttgart and Borussia Dortmund youngster has registered five goals and four assists in 49 games for the Blues this season.
The 29-year-old has also featured from the start on 30 occasions in the Premier League this term and now boasts a total of 12 goals and seven assists in 198 games for the Blues since his 2017 arrival.
Rudiger has helped Chelsea win the Champions League, Europa League, FA Cup, Club World Cup and UEFA Super Cup in that time and will hope to add a second FA Cup winners' medal to his collection when Tuchel's side meet Chelsea in this year's final next Saturday.
Reece James is also understood to be a target for Real boss Ancelotti, and the Blues have responded by making a new contract for the England international a priority once the club's takeover is completed.
Roman Abramovich reneging on his agreement to write off a £1.6m loan plunged the future of the club into fresh doubt earlier this week, but Chelsea are reportedly still confident that a takeover will go through. body check tags ::