Allen ended a 13-year spell with the Swans after completing a deal to link up with former manager Brendan Rodgers at Anfield.
He told Swansea's official website: "Having come through the ranks and been part of Swansea City for so long I have built up a massive attachment to the club and the fans.
"I've really enjoyed my time at the club and will always be eternally grateful to the fans for the support they've shown me over the years.
"I've always been ambitious and while it is hard to leave a club I love, this is a fantastic challenge for me that I had to take."
Allen made 129 league appearances during his Swansea career, helping the club finish in a credible 11th in last season's Premier League.