It's a minimum-level deal for Green, who will add versatility and depth to the Cavs' bench. The team has been looking during the free-agency period to add athletic players who can defend multiple positions.
Green had "active discussions" with LeBron James before making his decision to sign with the Cavs, sources said.
The agreement was first reported by USA Today Sports.
The team is hamstrung by being in the repeater tax bracket, meaning adding Green will cost the Cavs nearly $8 million in salary and tax for next season. Their team payroll is now more than $135 million with more than $46 million in tax.
Green averaged 9.2 points and 3.1 rebounds in 22 minutes per game for the Orlando Magic last season. He previously played for Oklahoma City, Boston, Memphis and the LA Clippers.
Green sat out the 2011-12 season with a heart condition. He will turn 31 in August.
He has averaged 13.5 points and 4.7 rebounds during his career. Green has appeared in 38 postseason games, most recently with the Clippers in 2015-16.
Meanwhile, the Cavs are an option for guard Jamal Crawford, who became a free agent after reaching a buyout agreement with the Atlanta Hawks on Friday. The Cavs have also signed free agents Kyle Korver and Jose Calderon since the start of free agency.
The Associated Press contributed to this report.