The Red Devils on Wednesday completed the transfer of the 21-year-old on a five-year contract.
United reportedly paid Swansea City £15 million for James' signature, plus an additional £3m in add-ons.
James becomes Ole Gunnar Solskjaer's first signing since taking over the reins on a permanent basis at the end of March.
The Welsh winger scored four goals and assisted a further nine in 33 Championship appearances for the Swans last season.
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"I’m very proud to join the biggest football club in the world," James tweeted. "This is a new and exciting chapter in my life and I can’t wait to get started. Thank you to everyone who has shown faith in me. Hard work starts now!"
He added: "This is one of the best days of my life. Every kid grows up wanting to play for Manchester United. It's the perfect place for me to develop."