The 26-year old has signed a four-year contract with the club.
Clucas's former club Chesterfield will earn around £760,000 from a sell-on clause.
"I've seen a lot of Swansea and one of the main reasons I decided to come here was because of the way they play football. I think it suits me." Clucas told the club's website.
"I had a bitter sweet season last year with relegation. Because of my playing background, I never thought I would play in the Premier League so I relished every game I played. From the first time I kicked a ball in the Premier League it was a dream come true."
"Once you've had a taste for the Premier League you always want to go back. So when the chance came up for me to do that again at Swansea, I grabbed the opportunity with both hands.''