Turner's contract is worth $3.278 million annually over two years, according to a report from Jeff Zillgitt of USA Today Sports.
Turner, 25, averaged 14.0 points, 5.0 rebounds and 3.2 assists over 81 games with the Philadelphia 76ers and Indiana Pacers last season. Turner shot 42.5 percent from the field and 32.1 percent from three-point range, while posting a 12.4 player efficiency rating and 1.4 win shares.
Indiana acquired the 6-foot-7, 205-pound Turner at the trade deadline deal that sent veteran forward Danny Granger (now with the Miami Heat) to Philadelphia. Turner's per-game scoring average with the 76ers was 17.4; with the Pacers it was 7.1.
His minutes were limited during the playoffs: Turner logged an average of 12.4 minutes in the Pacers first-round series against the Hawks, 14.0 in the conference semifinals against the Washington Wizards and 4.0 in the conference finals against the Heat.
Turner is now entering his fifth season in the league after being selected second overall in the 2010 draft out of Ohio State.
This offseason, the Celtics added the following players in trades: Forwards Dwight Powell and Erik Murphy and center Tyler Zeller. The team announced on Monday that it has waived guard John Lucas III and forward Malcolm Thomas.
The Celtics, who finished 25-57 in 2013-14, open the upcoming season Oct. 29 against the Brooklyn Nets.