The Portuguese winger has been linked with a United exit since the beginning of the season, and Juventus are now ready to sanction a £12m bid this summer.
The three-year deal on the table will double his United wages, making him one of the club's highest paid players.
Nani is out of contract at the end of next season, and has so far refused to pen an extension at Old Trafford. The report claims Juventus came close to taking the winger earlier last season, but the two parties couldn't agree terms over a price.
Paris St Germain were also reportedly chasing the former-Sporting star, and even had a £10m rejected.
But Juventus now lead the race, and a club source told The Sun: "We are following this one up. Nani is an interesting player and figures highly in our future projects."
The 26-year-old winger has found it difficult to break into the first-team this season.
It wasn't long ago that he was named Players' Player of the Year at Old Trafford, but just two seasons later and Nani has fallen down the pecking order, behind the likes of Ashley Young, Danny Welbeck, Shinji Kagawa and Antonio Valencia.