The Bulls announced the move Tuesday after reports surfaced over the weekend.
The 6-foot-9 Shengelia has appeared in 17 games for the Nets this season, averaging 1.5 points in just over eight minutes. The native of Tbilisi, Georgia, has played in 36 NBA games over two years.
Teague was drafted by the Bulls in the first round (29th overall) in the 2012 draft. In 68 regular-season games, he has averaged 2.1 points and 1.3 assists per game. Teague gives Brooklyn another option at point guard, where Deron Williams has been sitting out with ankle problems.