The 28-year-old is widely expected to leave Old Trafford this summer following a dismal spell in England. The Italian has only played 26 Premier League games over the past two seasons and made only 17 appearances in all competitions for Jose Mourinho's side last season.
And Napoli chairman Aurelio De Laurentiis has confirmed that I Partenopei have submitted a bid to United for the full-back's services.
Speaking to Radio Kiss Kiss Napoli, as cited by the Daily Mail, De Laurentiis said: "We have asked for Darmian on a loan with a right to sign this summer."
Mourinho has been aware of Darmian's desire to leave the Premier League club for a while, but a right back injury crisis at Old Trafford could mean the United manager is forced to hold on to the Italian defender for the time being.
Pre-season injuries to United captain Antonio Valencia and new signing Diogo Dalot means that Darmian is the only fit and available player in the right back slot ahead of the start of the Premier League season in 11 days' time.
It is understood that the 28-year-old's departure could be put on hold until United's injury woes ease up.
The Italy international has been a part of United's pre-season tour of America, featuring in all of the Red Devils' stateside games, and even wore the captain's armband for Mourinho's team in the friendly against AC Milan.
Manchester United kick off their Premier League season with a home tie against Leicester City on Friday 10 August.