Gibson joins former Bulls teammate Jimmy Butler, who was traded to the Timberwolves last month to reunite with Thibodeau.
The Oklahoma City Thunder acquired Gibson from the Bulls at the trade deadline last season.
In 23 games (16 starts) for the Thunder, the 6-foot-9, 236-pound Gibson averaged 9.0 points and 4.5 rebounds. Overall, he averaged 10.8 points and 6.2 rebounds in 78 games (71 starts) last season.
Gibson had his best season in 2013-14 with the Bulls, averaging 13.0 points and 6.8 rebounds in 82 games while finishing second in Sixth Man of the Year voting.
Specializing in defense and physicality, Gibson averaged 10.8 points and 6.2 rebounds in 78 games last season. He holds career averages of 9.4 points and 6.3 rebounds in 585 games.
Gibson expressed a desire to return to Oklahoma City, but he also was candid about wanting an extended contract as he enters the later years of his career.