The South Korean has signed a three-year contract and will move to the Liberty Stadium subject to a work permit and international clearance.
Ki said: "This is something I have dreamt about since I was a child - playing in the Premier League against the best players in the world.
"I can't wait to play my first game.
"When I look at Swansea I see a club that is growing and growing, and the style here is very attractive to me.
"Last year I watched them a lot in the Premier League and they did things very differently to a lot of the other teams.
"They wanted to keep the ball and pass it around, which I love to do.
"When they didn't have the ball they worked so hard to get it back quickly.
"It was very impressive, and I think I can grow as a player here and help the club move forward.
"The manager is another big attraction. I remember watching him when I was younger and I was amazed by how good he was.
"Now I want to learn from him in order to help me become a better player. I learned a lot at Celtic - it helped me become more of an all-round player."