The 34-year-old was linked with Scunthorpe after his release from Swindon in May.
"Tommy brings a wealth of experience that is needed at this level," Bury manager Kevin Blackwell said.
"He is excellent on the ball and is one of the key players I have been trying to bring to the club. He had offers at higher levels but he was very impressed with the new set up here at Gigg Lane. He will be a great asset."