AS says talks have been held between the two clubs over the Argentina international, who is open to a move to the Spanish giants.
In recent weeks there has been contact between Chelsea and Real Madrid. During these talks, the Blues enquired about the situation of several members of Real Madrid's squad.
And the Madrid club took the opportunity to ask about San Martín native Enzo.
After these preliminary meetings, Real management now knows that his signing could be made for around €120m, an amount that could only be paid if a major sale occurs first.
Enzo is 'interested' in the move - and he also likes the city. In a small group meeting, Enzo has made it clear: he loves Madrid, where he escapes when Chelsea gives the squad days off.
At the moment, nothing is formal, but Enzo is now warming to the prospect of a transfer.