As Football Italia reports with reference to Sky Sport, the Italian club has already taken the first steps in order to agree with Barcelona on the loan of the 21-year-old player during the winter transfer window.
In September, head coach of the blue garnet Ronald Koeman told the midfielder that he was not counting on him this season.
Despite this, the footballer expressed his intention to stay in the team.
Puig has played just four Barcelona games this season, playing a total of 77 minutes.
The young midfielder's contract runs until July 2021 with the option to extend it for another two seasons.
Earlier, Messi spoke about the possible return of Neymar to Barcelona.