The 28-year-old has been in and out of the lineup this season, with groin and back injuries limiting him to just 24 games. He has thrived when healthy, however, averaging 19.1 points per contest, which would be a career high.
Fournier has averaged at least 15 points per game in each of the last five seasons and is a career 37.6 percent shooter from three-point range.
The wing also showcased his offensive upside with 31 points in Friday's win over the Brooklyn Nets, shooting 10-of-13 from the field and 6-of-8 from three-point range.
He would be a welcome addition for Boston, which needs more scoring options behind Jaylen Brown and Jayson Tatum. Guards Kemba Walker and Marcus Smart have returned from injuries but neither has been consistent offensively this season.
The Celtics still have high expectations but enter Monday just 21-21 and in sixth in the Eastern Conference.
It could force the front office to be aggressive ahead of the March 25 trade deadline seeking help to keep up with the Philadelphia 76ers and Brooklyn Nets.
Fournier is set to become a free agent in the offseason after making $17 million in 2020-21.