James signs on a five-year contract, with an option to extend for a further year.
The 21-year-old Wales international made his professional debut for Swansea City in February 2018 and his senior Wales debut in November 2018, having previously represented the nation at various youth levels.
United manager Ole Gunnar 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.