Thibodeau served as Noah's head coach with the Chicago Bulls for five seasons from 2010-11 through 2014-15.
The 33-year-old Noah spent the first nine seasons of his career with the Bulls before signing with the Knicks.
Injuries limited him to just 53 games in New York over two campaigns. Last season, Noah averaged career lows with 1.7 points and 2.0 rebounds in seven contests.
Noah's most productive seasons came under Thibodeau in Chicago.
He was named an All-Star in both 2012-13 and 2013-14, and he won the 2013-14 NBA Defensive Player of the Year award after a career year saw him average 12.6 points, 11.3 rebounds, 5.4 assists, 1.5 blocks and 1.2 steals per game.
Since taking over as Minnesota's head coach prior to the 2016-17 season, Thibodeau has made a habit out of signing and trading for his former Bulls players.
The T-Wolves traded for Jimmy Butler and Taj Gibson last offseason before signing Derrick Rose and Luol Deng this offseason.
While Butler helped the Timberwolves reach the playoffs last season for the first time since 2003-04, he grew unhappy with the organization and requested a trade three weeks ago.
Despite his affinity for former Bulls, Thibodeau has little need for Noah due to Minnesota's depth at center, which features Karl-Anthony Towns, Gorgui Dieng and Justin Patton.