The 26-year-old has been heavily linked with a move away from the Premier League champions this summer, after falling way down the first-team pecking order in the past 12 months.
Antonio Conte is a long-term admirer of the Portugal international, and is ready to offer him an escape-route from Old Trafford, by giving him the chance to continue his career in Turin.
However, Juventus could face competition from AS Monaco, who are also keen on signing Nani ahead of the 2013/14 campaign.
The free-spending French club are keen to build a squad capable of challenging Paris Saint-Germain for the Ligue 1 title next year, following their promotion back to the top-flight under Claudio Ranieri in 2012/13.
Sir Alex Ferguson was reluctant to let Nani leave last season, but since his retirement new manager David Moyes might sanction the attacking midfielder's sale, now that he has the final say on incoming and outgoing players at Manchester United.
Italian publication Corriere dello Sport claims that Nani does not feature in the former Everton boss' plans moving forward, and so will be allowed to leave England - for France or Italy - later this summer.