The Irishman, 37, succeeds Paulo Sousa - who quit to take charge of Leicester City - at the Welsh club.
"It's a very proud moment for me. I've very proud to become manager of Swansea City - a progressive club that plays the game the right way," Rodgers told Swansea's official website.
"My aim is continue that philosophy and help the club progress. I want to play offensive, attacking football, but with discipline.
"I will inherit a wonderful group of players and, if I can help them develop and improve, I will be very pleased. I've seen the passion of the Swansea fans for myself when I was here last season and I can assure them I've come here to be a winner."
Barnsley have given a trial to their former striker Michael Boulding, who has been released by the League Two side Bradford CIty.Boulding was a player at Oakwell for two years from 2004, scoring 10 goals in 27 starts,The Hull City midfielder James Ha
Boulding was a player at Oakwell for two years from 2004, scoring 10 goals in 27 starts.
The Hull City midfielder James Harper will miss the start if the season after suffering a broken foot on his debut.
Harper, signed from Sheffield United, broke a metatarsal in his right foot during Wednesday's 6-0 win at North Ferriby United.
Norwich City have completed the signing of Gillingham striker Simeon Jackson for an undisclosed transfer fee.
He has signed a two-year deal at Carrow Road while Cody McDonald has gome the other way, joining the Gills on a season-long loan.
Preston North End have signed the full-back David Gray on a free transfer from Manchster United. Gray, 22, has signed a two-year contract at Deepdale.