The News of the World says the defender, 29, is one of those said to have been most affected by the departure of Jose Mourinho.
He had a close relationship with the Blues' former manager, after patching up their friendship following a row a couple of seasons ago.
Mourinho persuaded him to sign a five-year deal towards the end of last season - a contract that would see him end his career in the Premier League. But with his mentor gone, Carvalho could be open to a move and Real are getting ready to test Chelsea's resolve.
The length of the player's contract means the Spanish giants will have to put in a serious offer to tempt the Londoners into selling.
And that would probably mean a figure in the region of £10million.