This is according to reports cited by The Daily Express, which claim that the reigning European champions are also still hoping to include German international Sami Khedira as part of any deal.
Chelsea - who bolstered their midfield unit considerably with the signing of Cesc Fabregas yesterday - are not believed to be particularly tempted by the offer of Khedira, however, and it appears very much as if Real Madrid are going to have to considerably up the stakes in order to get their man.
A tenacious, tireless and energetic midfield battler, £25 million-rated Ramires has played a crucial role for Chelsea over recent years, although it appears that his habit of collecting cards and launching into one or two reckless challenges has started to wear somewhat thin with a number of the Stamford Bridge faithful.
An apparent reluctance to take on Khedira is a little surprising, particularly considering that Jose Mourinho was responsible for taking him from the Bundesliga to the Santiago Bernabeu in the first place.
Khedira - who is currently in Brazil as part of Joachim Low's star-studded Germany squad for the World Cup - last year labelled himself as one of Mourinho's 'disciples' in the Spanish capital, leading to speculation that he was not entirely happy under Carlo Ancelotti.
Just why Chelsea are so unreceptive to the prospect of signing Khedira is unclear, although one would certainly understand if they were to have reservations over his long-term fitness after the 27-year-old missed most of last season with a serious knee injury sustained during November's friendly clash with Italy.
If reports are to be believed, it is not Khedira that Chelsea are looking to receive in any potential transaction, but rather Real Madrid teammate Luka Modric.
Modric - who featured for Croatia last night during their controversial defeat to Ramires and Brazil in the opening game of this summer's keenly-anticipated showpiece - has grown into an influential figure at Real under Ancelotti, however, and it seems unlikely that they will be willing to part with him without a clear incentive to do so.
With the respective sides having seemingly reached an impasse, then, it appears that someone will have to alter their expectations moving forward or Ramires will be staying at Chelsea for the foreseeable future.