Juventus were keen on landing Danilo at one stage, but Serie A rules stipulate that Italian clubs can only sign one none-EU national and the reigning champions appear to be closing in on a deal to take Douglas Costa from Bayern Munich.
Antonio Conte likes Danilo and is keen for Chelsea to make an attempt to bring him on board ahead of the new season, but Real Madrid would need to find a suitable replacement before selling the Brazil international.
Danilo endured some testing moments at the Santiago Bernabeu in 2016-17, but the Madrid decision-makers believe that he can still come good under Zinedine Zidane.
Madrid signed Danilo from Porto for £28 million in 2015 and they would not consider any offers below what they originally paid, despite his indifferent time at the club.
Chelsea face the need to strengthen in both wing-back positions as they struggled in games that Marcos Alonso and Victor Moses missed through injury last season.
Juventus' Alex Sandro remains the main target on the left hand side, but Chelsea have only made slow progress in negotiations so far.
Chelsea are also expected to make an approach for Danilo's team-mate Alvaro Morata in the coming weeks.
The Spain international striker has been heavily linked with a move to Manchester United this summer, but the Red Devils have tied up a £75 million deal for Romelu Lukaku after failing to make headway in discussions with Madrid.