Spanish daily AS reports that Blues owner Roman Abramovich has offered 35million euros for the Br azilian, a player they claim new Chelsea boss Luiz Felipe Scolari has had his eye on for some time.
Real need to raise cash for a possible bid for Cristiano Ronaldo, and Robinho is thought to be the favourite in their squad to be sacrificed.
Should Ronaldo arrive at the Bernabeu, Robinho would most likely find himself on the bench for much of the season, so could be open to a move.
Scolari officially starts work at Stamford Bridge on July 1, and is seemingly already planning the anticipated overhaul of his squad.
Frank Lampard has been subject of repeated interest from old boss Jose Mourinho at Inter Milan, while Didier Drogba has made no secret of his desire for a move away from London.