A former fourth-overall pick, Thompson spent the first nine seasons of his NBA career with the Cleveland Cavaliers.
The Toronto, Ont. native has averaged 9.4 points and 8.7 rebounds per game throughout his career, including a personal-best 12 points per game last season.
Thompson figures to shore up the interior weaknesses of Boston's starting five, offering post-scoring and rebounding to a club that features perimeter stars like Jayson Tatum and Kemba Walker.
His addition to the Celtics comes just hours after former All-Star Gordon Hayward decided to leave the team in favour of joining the Charlotte Hornets.