The Welsh international's transfer was agreed in principle last week, but on Wednesday afternoon, James was officially unveiled as a United player.
He told MUTV: "For me, it's surreal, really. I think every kid grows up wanting to play for Manchester United. I never thought I'd be able to play against them, never mind play for them, so it's surreal, and I'm really looking forward to it. I can't wait to get started.
"Last season I didn't think I was going to get much first-team football. I remember going into the office at the start of this year and saying I wanted to go on loan, so I got told that I'd be a part of something. Obviously, you go out of the office thinking, 'Is that true?', and to get more opportunities this year has been amazing. It's been a great six months for me – to get that many games for Swansea and to obviously get my Welsh debut as well has been amazing."
On manager Ole Gunnar Solskjaer, he added: "Yeah, I've spoke to him since I've been here. I think the ambition they've got for the club is amazing for the next few years – to get players in who are hungry and ready to work hard. I believe [in] that and I'm one of those players. I'm willing to work hard. It's a challenge I'm looking forward to."