The 21-year-old has signed a five-year contract at Old Trafford, with the option to extend by a further year, to become manager Ole Gunnar Solskjaer’s first signing in charge of the Red Devils.
James, a graduate of the Swansea academy, scored five goals in 38 appearances during an impressive 2018/19 campaign with the Championship side.
Solskjaer said: “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 has impressed at Swansea since breaking into the first team following their relegation from the Premier League a year ago.
A potential move to Leeds collapsed in January and he was also linked with Brighton but United became favourites to land him in a £15million deal this summer.
The transfer is completed less than a month after the death of James’ father.
“This is one of the best days of my life and a challenge I am really looking forward to,” the young winger added. “The Premier League is the best league in the world and Manchester United is the perfect place for me to continue to develop as a player.
“This is an incredibly proud moment for me and my family, however, it is tinged with sadness that my father isn’t able to share this experience with us.
“I am looking forward to working with Ole, meeting the team and getting started.”
Swansea chairman Trevor Birch wrote in a statement: “We are obviously hugely disappointed to lose a player of Daniel’s quality, especially after such a fantastic breakthrough season for us and the enjoyment he gave to those watching him.
“However, given the level of those performances and the fact he has one year remaining on his contract, it was inevitable that once a club of the calibre of Manchester United came calling it would be virtually impossible to hold on to him.”