The Italian right-back has struggled for game time at Old Trafford, making just four Premier League appearances under Jose Mourinho this season.
Juve could offer him a way out of United, however, with the Serie A giants keen to replace 34-year-old defender Stephan Lichtsteiner, who is set to leave the club when his contract expires this summer.
United paid around £13 million to sign the defender in 2015, under the management of Louis van Gaal, and he has made a total of 80 appearances for the club, despite falling down the pecking order due to the form of Antonio Valencia.
United have not yet placed a value on the 28-year-old, though, and it remains to be seen if Juve will be able to negotiate a deal at the end of the season.
Darmian has particularly struggled for regular football since Mourinho replaced Van Gaal as United manager in 2016, but insisted back in October that he didn't regret joining the club.
"I made a choice [to come to Manchester United] and I don't regret it," Darmian said.
"Of course it's normal that any player wants to play as much as possible. I work hard every week and I'll keep doing it.
"I don't always watch Italian football and I don't know what will happen in the future, maybe there will be a chance to come back one day."
But opportunities have become even more reduced, with Darmian's sole appearance in 2018 coming in the FA Cup fourth-round win at Yeovil Town.