However, the 60-year-old Australian led Spurs to a historic Europa League victory at the end of the season to end the club's long trophy drought.
Postecoglou will be presented as Forest boss at a news conference at 13:30 BST on Thursday.
Forest owner Evangelos Marinakis said: "We are bringing a coach to the club who has a proven and consistent record of winning trophies.
"His experience of coaching teams at the highest level, along with his desire to build something special with us at Forest, makes him a fantastic person to help us on our journey and achieve consistently all our ambitions."
Forest sit 10th in the Premier League table after taking four points from their opening three games of the season.
Nuno's last match in charge was a 3-0 defeat by West Ham before September's international break.
The 51-year-old Portuguese coach took charge of Forest in December 2023 after the dismissal of Steve Cooper and helped preserve their top-flight status.
Last term he guided the club to seventh in the Premier League - their highest finish since 1994-95 - as they qualified for Europe for the first time in three decades.
However, Nuno's position had been uncertain for two weeks since he publicly declared his relationship with Marinakis had deteriorated, while there were also suggestions of a rift with Forest's global sporting director Edu.
"After gaining promotion to the Premier League, then building consistently season after season to secure European football, we now must take the right step to compete with the very best and challenge for trophies," Marinakis said.
"Ange has the credentials and the track record to do this, and we are excited he is joining us on our ambitious journey."
BBC Sport understands Forest's players have been caught by surprise by Nuno's departure.
Many of them are on international duty and not due back for training until later this week.
But word spread, with some players hearing the news from other members of the squad.
Forest winger Callum Hudson-Odoi, who signed a new three-year deal with the club less than two weeks ago, thanked the outgoing manager in a post on X.
"I learned a lot from you and made some great memories along the way. I wish you nothing but the best for the future in your new chapter," Hudson-Odoi wrote.
Postecoglou's first game in charge will be Forest's trip to Arsenal on Saturday.
He becomes the eighth permanent manager since Marinakis took over in May 2017.