The 24-year-old is reunited with Shakers boss David Flitcroft, who he worked with at Barnsley.
Flitcroft said of the former Chelsea midfielder: “I have brought Jacob to the club as I feel he fits perfectly into the vision we have for our team on the pitch and into the new DNA profile for a Bury player.
“Jacob has not yet reached his potential and it is our aim and desire to give him a home where he can fulfil that.
“I believe that he will create opportunities and score goals. He can improve us in all forward areas of the pitch with his creation and execution quality.
“I worked with Jacob at Barnsley where he was an instrumental player in helping us stay in the Championship. In the time I worked with him we built a very good relationship based on trust, and I believe that this was a major factor in Jacob coming to Bury.”