The news of Green's signing was first reported by the Sporting News, while Zeller's news was first reported by the Boston Herald.
Green is expected to ink a one-year deal worth $1.4 million with the team that originally selected him with the No. 18 pick in the 2005 draft. He spent two seasons with Boston before being dealt to the Minnesota Timberwolves as part of the Kevin Garnett deal in 2007.
Zeller has agreed to a two-year, $16 million deal with the second year being a team option, a source told ESPN's Ramona Shelburne.
By signing Zeller and Green, the Celtics appear content to head into the season with the roster they have constructed. Boston had been maintaining its salary cap flexibility while examining other trade and free agent options
The 30-year-old Green averaged 8.9 points per game in 69 appearances for the Heat last season, including 14 games as a starter. An erratic season for Green saw the nine-year veteran go from being a high-scoring sixth man for the Heat in the first half of the season to barely playing over long stretches entering the playoffs. Despite those inconsistencies with his role, Green repeatedly said he hoped to remain in Miami long term.
Stability, however, has evaded Green for most of an NBA career in which he's played for the Celtics, Houston Rockets, Timberwolves, Dallas Mavericks, Brooklyn Nets, Indiana Pacers and Phoenix Suns. He signed a one-year deal for the veterans minimum with the Heat last summer.
Green has shot 36.1 percent beyond the 3-point arc for his career. During the 2013-14 season with Phoenix, Green shot 40 percent while hoisting a career-high 510 attempts.
The 26-year-old Zeller, the 17th overall pick in the 2012 draft, appeared in only 60 games for Boston this season -- a year after starting 59. Zeller was the odd man out in the Celtics' log-jammed frontcourt and logged 25 DNPs, including three in the postseason.
The Celtics acquired the 7-foot Zeller -- and the first-round pick that would help land Isaiah Thomas -- as part of a three-way trade in July 2014 that helped the Cleveland Cavaliers clear the necessary cap space to bring back LeBron James.
Zeller averaged 6.1 points and 3 rebounds over 11.8 minutes per game last season. He earned $2.6 million before reaching restricted free agency. The Celtics extended a $3.7 million qualifying offer in June.
Information from ESPN's Michael Wallace was used in this report.