The Liverpool midfielder spent last season on loan at the Trotters, earning the player of the year award.
"Jay was my top target," manager Dougie Freedman told the BBC. "He's got the potential to be a very, very good player.
"What I like about him is that he comes in every day with the attitude to get better. At my last club that's what it was all about - the younger players coming through the system and making sure they apply themselves in the right way.
"That's what struck me about Jay - not only was he a fantastic football player with a great pedigree coming from Liverpool, but his hunger and desire to get better was really the thing that struck me."
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