Lodi is the newly-promoted side's 18th signing of the summer.
The 24-year-old can join Steve Cooper's side on a permanent deal for £25m at the end of the season.
Lodi spent three seasons with Atletico Madrid after joining from Brazilian side Atletico Paranaense in the summer of 2019.
He made 118 appearances for Diego Simeone's side, helping them win the La Liga title in the 2020/2021 season.
Lodi scored the only goal of the game when Atletico Madrid knocked Manchester United out of the UEFA Champions League at the round of 16 stage at Old Trafford last season.
Some of Nottingham Forest's other summer recruits include Jesse Lingard, Dean Henderson, Taiwo Awoniyi, Omar Richards, Moussa Niakhate and Morgan Gibbs-White.
Nottingham Forest have started their season with one win, one draw and two losses so far.
Lodi, who has won 15 caps for the Brazil national team, could make his debut at Manchester City on Wednesday.
Meanwhile, Atletico Madrid are set to sign Tottenham Hotspurs' Spanish left back Sergio Reguillon as Lodi's replacement.