James, the Wales winger, makes the step up from the Championship in a deal worth a reported £18million, after a stunning breakthrough season in Wales.
He will officially join United when the international transfer window opens next week, having undergone a medical with the Red Devils on Thursday.
“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,” a statement read.
“Daniel has successfully completed a medical at the Aon Training Complex.
“Further details will be communicated once the international transfer window opens next week.”
James, who has won two caps for Wales, left their international base near Chester on Thursday to undergo his medical at United’s Carrington training ground.
He is expected to feature in the Dragons’ Euro 2020 qualifiers against Croatia, on Saturday, and Hungary next Tuesday.
A host of Premier League clubs had been linked with James, who was close to joining Leeds in January before the move collapsed at the 11th hour.
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He becomes United’s first signing of the summer transfer window, and also the first signing made by Ole Gunnar Solskjaer since his appointment as the club’s manager on a permanent basis last December.
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