This is according to reports from German publication Bild cited by talkSPORT, with Khedira having fallen in the midfield pecking order at the Santiago Bernabeu recently thanks to excellent form shown by the likes of Luka Modric and Isco.
Indeed, Khedira - who last week was named by Joachim Loew in Germany's provisional 30-man squad for the 2014 World Cup - has endured a torrid season in La Liga, only recently returning from a devastating knee injury suffered while on international duty last year.
The news that Real Madrid apparently appear content to ponder such reasonable offers for Khedira is likely to prove a significant boost for Chelsea, with the Premier League giants having been linked with a summer swoop for the 27-year-old at a much higher price of £36 million.
A former favourite of current Chelsea boss Jose Mourinho during the latter's own spell at Real Madrid, Khedira would be a fabulous addition to the squad at Stamford Bridge and would certainly provide an instant upgrade over the likes of John Obi Mikel.
Chelsea have not been shy in indulging in a spot of heavy transfer spending since Roman Abramovich's arrival at the club over a decade ago, so a figure of just over £20 million would presumably be well within their price range.
The only point Chelsea and Mourinho must ponder is just whether they are willing to take the risk on a player who has missed so much first-team action this season.
Although with the likes of Serie A duo Fiorentina and Roma also believed to be interested in his signature, they will presumably have to decide sooner rather than later.