In return, the Raptors will acquire Marcus Camby, Steve Novak and a first-round draft pick in 2016.
The reports also indicate that a future second-round pick will be included in the deal, but some sources say there will be two second rounders involved in the transaction.
The New York Times was the first outlet to report the deal. Sportsnet, Yahoo! Sports and CBS also confirmed the deal independently.
Bargnani was drafted first overall by Toronto in 2006.
The seven-foot, 256-pound Italian forward has averaged 15.2 points, 4.8 rebounds and 1.3 assists per game over seven seasons with the Raptors.
Bargnani scored 12.7 points per game last season with 3.7 rebounds and 1.1 assists over 35 games. He missed most of the season with two significant elbow injuries.