Coach Antonio Conte wants Danilo to compete with Victor Moses for the right wing back position and the Brazil international is prepared to leave the Bernabeu, the Evening Standard has claimed.
Chelsea are still building their squad for next season.
The Blues have signed Willy Caballero and Antonio Rudiger but they are eager to add more quality as soon as possible.
They have made an initial bid of £28million to sign the Brazilian.
And club owner Roman Abramovich is not prepared to sanction an increased bid to rush through a move.
The Spanish giants want £30.8million for the South American but Chelsea are unwilling to meet the fee.
Conte's side are preparing for the new season with the Italian unhappy with how things have moved in the transfer market.
He is hoping to add at least three new players before the end of the month.
