The midfielder was at Anfield on Thursday to watch the Reds' 3-0 Europa League third qualifying round second leg win over FC Gomel.
And, after Reds manager Brendan Rodgers revealed the move was "imminent", the 22-year-old has now passed a medical to seal his switch to the Merseyside giants.
Allen told his new club's official website, www.liverpoolfc.com: "I feel incredible.
"Everyone knows the history of this club, it's a massive club, and I'm delighted to have joined.
"The passion that people here have for football is something I share and I want to be part of that.
"I'm looking forward to being part of some great years ahead for Liverpool."
Rodgers said: "I'm absolutely delighted Joe has made the decision to come with us on this journey.
"He is a player whose profile will fit perfectly with the ideas of this group.
"His ability to control and dominate the ball is an important ingredient in our attempt to gain success on the field.
"Joe has had a fantastic education at Swansea and will now begin the next chapter in his exciting career, and I wish him all the very best in what I'm sure will be a long and distinguished career here at Liverpool."
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