The 21-year-old spent three years at Anfield after joining from Charlton Athletic in May 2010.
A message on Liverpool's official website read: "Everybody at Liverpool FC would like to wish Jonjo all the best for the future."
Meanwhile, the Swans have revealed that the one-capped England international has cost them £5m and he has signed a four-year contract at the Liberty Stadium. There are also potential add-ons to the transfer fee depending on success.
He made 69 appearances for the Merseysiders, scoring seven goals in the process.