The 29-year-old has moved to the League Two club after finding first-team opportunities limited at Prenton Park.
Shakers manager David Flitcroft has said that he is pleased to team up with Akpa Akpro again, having previously played with him at Rochdale.
He told Bury's official website: "Jean-Louis is obviously a player I have worked with before.
"He is a player that loves football and loves to be involved in football. He's got a fantastic commitment to his game and has that pace, power and strength that supporters love to see.
"He's a player we have been tracking for a while and with him being out of the side at Tranmere we are delighted to have his services with us until the end of the season."
Akpa Akpro joined Tranmere from Rochdale in June 2012.