Juve sporting director Fabio Paratici has met with an intermediary of the 25-year-old Brazilian, with a view to completing a deal before the 2017-18 season.
The Serie A champions have made an initial offer of €15 million plus bonuses, but Madrid is hoping to secure a fee of €30m for the right-back.
Wtih Dani Alves confirming his Juventus departure on Thursday, the Serie A champions are in need of reinforcements at the right-back position.
Negotiations will likely continue for Danilo, who has failed to lock down a starting role since arriving from Porto in a €31.5m deal in the summer of 2015.
With Dani Carvajal secure in his spot as Real Madrid's first-choice right-back, the Blancos could look to recoup some of their investment in Danilo after just two seasons.
Sources close to the defender say that he is still happy in Madrid, however, and is not necessarily looking to move on from the Bernabeu.