A nagging right hamstring injury limited Addai to 12 games in 2011 after he missed eight games the year before with a nerve injury in his left shoulder.
Addai, who turned 29 on Wednesday, isn't a lock for a roster spot, although he offers a veteran presence behind second-year backs Shane Vereen (2011 second-round choice) and Stevan Ridley (2011 third-round choice), as well as "passing back" Danny Woodhead.
Addai rushed for 4,453 yards and 39 touchdowns over six seasons in Indianapolis, making the Pro Bowl in 2007.
Addai was a first-round draft choice of the Colts in 2006, selected No. 30 overall. The Patriots selected running back Laurence Maroney with the 21st pick that year, although they also liked Addai.
Information from The Associated Press contributed to this report.