Theis will compete for minutes in the frontcourt, including at backup center. Boston lost five-time All-Star Al Horford in free agency, but signed free agent center Enes Kanter to a two-year deal and will return 2018 first-round pick Robert Williams.
Theis, 27, has averaged 5.5 points and 3.9 rebounds in his two seasons with the Celtics.
Theis, a four-time German All-Star, played two seasons with the Brose Bamberg of the EuroLeague.
Brad Wanamaker agreed to a one-year deal to return to the Celtics too, his agent Steve McCaskill told ESPN. Wanamkaer, 29, played six seasons in Europe before joining the Celtics a season ago. He was the NBA's oldest rookie at 29 years old.