The Daily Express reports that the Italian side are keen to force through a deal to sign the 27-year-old.
Nani only recently signed a new five-year contract with United and has featured more regularly for the club this season.
Monaco are also reportedly interested in the highly-rated winger.
Tuttosport reports that Juventus boss Antonio Conte is a fan of Nani and is keen to bring in the winger in January.
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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
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