The Bulls announced the move Wednesday in a release promoting a Thursday news conference.
Forman has been director of player personnel the past five years. He joined the organization as a scout in 1998 and became the team's director of player personnel in January 2004.
Paxson is expected to retain a major say in personnel moves, but was not immediately available for comment Wednesday. He had been GM since April 2003, when he was officially hired as executive vice president of basketball operations.
The new titles reflect a shift that had already taken place over the past year, with Forman assuming more day-to-day responsibilities. That included the lead role in contract negotiations with Luol Deng and Ben Gordon and input in the coaching search last summer.
Besides helping Paxson evaluate potential trades, free agents and coaches, Forman coordinated Chicago's amateur and pro scouting as director of player personnel.
He joined the Bulls scouting department in 1998 after serving as an assistant coach and recruiting coordinator at Iowa State and New Mexico State the previous 10 years, helping the Cyclones win a conference championship and the Aggies reach the 1992 round of 16.
The move comes after a season in which the Boston Celtics eliminated the Bulls from the playoffs in seven games in the first round.
Chicago fans had been particularly critical of Paxson, who played on three Bulls championship winners, as the team stumbled into the lottery last year and struggled for much of this season despite the arrival of rookie of the year Derrick Rose.
They even booed him during a ceremony honoring longtime analyst Johnny "Red" Kerr, who died shortly afterward, and around that time, came reports that Paxson would be out either immediately or at the end of the season.
After about a week, he responded by saying, "I'm the GM of the Bulls. I'm not resigning. I'm here and I'm going to be here, and that puts an end to it."
The Bulls won 12 of 16 down the stretch to finish 41-41 and grab the seventh seed in the Eastern Conference, thanks to Rose and veterans such as Gordon, the trade with Sacramento for John Salmons and Brad Miller, and the improved play of Joakim Noah and Tyrus Thomas.
That led to one of the most dramatic playoff series in league history and a steady, deafening roar from an re-energized fan base.
It was a sharp turnaround for a franchise that appeared to be in disarray.
Chairman Jerry Reinsdorf even called the season "a disaster" and "embarrassing" during an interview with Comcast SportsNet at one point and said he'd been tempted to stand up and boo, but he also made it clear he didn't blame Paxson.
"If there's one person that is not responsible for what's going on right now, it's John Paxson," Reinsdorf said in that interview. "I have tremendous confidence in John Paxson."
Paxson played on the Bulls' first three championship teams in the 1990s and delivered a memorable shot during the 1993 finals against Phoenix, making the series-winning 3-pointer with 3.9 seconds left.
He later served as an assistant coach and broadcaster before replacing Jerry Krause as the general manager.
Paxson drafted Kirk Hinrich with the seventh pick in 2003, took Ben Gordon with the third pick in 2004 and acquired the rights to Deng from Phoenix. They formed the core that led the Bulls to the playoffs in 2005, '06 and '07 before the collapse last season.