The Bernabeu and Blues owner Roman Abramovich wants to burst onto the scene with a major coup and sees Ramos as the player to do just that.
The transfer fee will be coupled with a five-year contract for Ramos and will worth £8.2m (over $12m).
Chelsea has not been able to make many of their priority transfer targets with the latest failure being that of young Brazilian talent Neymar.
The deal involving Ramos would be for a five-year contract worth £8.2m (over $12m) per season but Chelsea will only have about a week left to tie up any agreement.
The London club’s only additions this summer are midfielders Yossi Benayoun from Liverpool and Brazilian international Ramires, who joined from Benfica, while the likes of Ricardo Carvalho and Michael Ballack have moved on.
However, the big money offer from Chelsea could yet “sweeten” the negotiations as Real would find the extra funds handy as they continue to work on bringing in reinforcements in other areas.