"This is one of the best days of my life and a challenge I am really looking forward to," James said in a statement.
"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."
James, who joined Swansea from Hull City in 2014, scored five goals in 38 matches in all competitions last season as they finished 10th in the English second tier.
He scored on his first Wales start against Slovakia in their Euro 2020 qualifier in March.
James' arrival marks manager Ole Gunnar Solskjaer's first signing as part of his squad overhaul in the close-season following a disappointing sixth-placed finish in the Premier League, 32 points behind champions Manchester City.
"Daniel is an exciting young winger with lots of skills, vision, exceptional pace and a good work ethic," Solskjaer said.
"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."
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
