The 30-year-old Green agreed to a deal last week with the defending Eastern Conference champions. The one-year pact is worth a reported $2.3 million, according to ESPN.
Green had "active conversations" with Cavaliers star LeBron James before making his decision to sign with the team, per ESPN.
The minimum-level deal will cost the Cavaliers roughly $8 million in salary and tax for next season. Cleveland's current payroll for the 2017-18 season is $135 million with $46 million in tax.
Green averaged a career-low 9.2 points on 39.4 percent shooting and 3.1 rebounds in 69 games with the Orlando Magic last season.
The former No. 5 overall draft pick in 2007 has career averages of 13.5 points and 4.7 boards over 705 games.