The Spanish giants have been watching the Blues ace for some time, with Carlo Ancelotti known to be a big fan of his abilities.
Real were linked with a summer move for the Brazilian, but a move failed to materialise.
But, Ancelotti is ready to revive his interest in the New Year with a £20m bid.
Chelsea are desperate to keep Ramires, but the La Liga side are thought to be ready to further sweeten the deal by offering players in exchange.
Xabi Alonso and Sami Khedira have both been linked with the London club and it is thought that Real may be willing to allow one of them to head to Stamford Bridge as part of a potential agreement.
Alonso, 31, is set to be a free agent once his contract expires at the end of the campaign, while his fellow midfielder is rumoured to be frustrated with life at the Bernabeu.
Ancelotti would prefer a straight cash move, but his desire to land Ramires may allow Jose Mourinho to negotiate a good deal.
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