It is understood the 21-year-old has signed a five-year contract with the Old Trafford giants and has become Ole Gunnar Solskjaer’s first addition since his arrival as manager in December 2018.
The pacey winger lit up the Championship in 2018-19, scoring five goals and assisting 10 more in 38 appearances in all competitions.
James also made a huge impact in this year's FA Cup, scoring a memorable individual goal against Brentford in the fifth round, before putting in a stellar display as Swansea succumbed to a narrow defeat against Manchester City in the quarter-finals.
As a result of his excellent form, he was also handed his first two caps for Wales by United great Ryan Giggs, scoring in his second appearance against Slovakia in March.
United have released an official statement confirming James's arrival via their club website on Friday, which reads: "Manchester United is delighted to announce that it has agreed, in principle, terms with both Swansea City and Daniel James for his transfer to the club
"Daniel has successfully completed a medical at the Aon Training Complex. Further details will be communicated once the international transfer window opens next week."
A native of Beverley in Yorkshire, James came through the youth system of nearby Hull City before being snapped up by Swansea for a nominal compensation fee of £72,000 in 2014.
He came close to joining Leeds United during the January transfer window, and despite conducting interviews with the in-house television channel at Elland Road, a deal could not be closed before the deadline.He will now join up with Solskjaer and the United squad for the first time on July 1 when the players report for pre-season training ahead of their summer tour to Australia and Asia.