According to the club's official website, the forward joins on a five-year contract, subject to international clearance.
The club welcomed the 21-year-old on Twitter:
James' deal contains an additional 12-month option if United wish to retain the exciting winger.
Per the club's website, United manager Ole Gunnar Solskjaer expressed his delight at the signing:
"Daniel is an exciting young winger with lots of skills, vision, exceptional pace and a good work ethic. He had a great season with Swansea City and has all the attributes needed to become a Manchester United player.
"We are delighted he has signed with our club and we are all looking forward to working with him. This is the perfect environment for Daniel to continue his development."
James tweeted a picture of himself wearing United's new shirt, and he expressed his pride at moving to the Red Devils:
The youngster has been one of the hottest prospects in the EFL Championship, but the step up to the Premier League could take him time.
James has the attributes to develop into one of United's most dynamic players after scoring four goals and providing seven assists for Swansea last term.
The player could now form a lethal front line with Marcus Rashford and Anthony Martial, with James able to play in all three attacking roles in a 4-3-3 formation.