ESPN's Brian Windhorst reported Thursday the Cavs and Spurs have had discussions involving Green, but the Spurs currently have a high asking price.
The Spurs continue to search for ways to clear cap space this offseason. Jabari Young of the San Antonio Express-News reported the team is exploring the trade market for LaMarcus Aldridge.
ESPN's Ramona Shelburne reported the Spurs have strong interest in signing Chris Paul or Derrick Rose, if they can free up enough money to make a deal work.
The Cavs have been linked to multiple high-profile players already this offseason, though Windhorst reported (h/t ESPN Cleveland) Thursday potential deals for Paul George of the Indiana Pacers or Jimmy Butler of the Chicago Bulls are unlikely because neither team is interested in a deal centered around Kevin Love.
Green has one more guaranteed year on his contract worth $10 million, per Spotrac. The 30-year-old, who started his career with the Cavaliers in 2009-10, averaged 7.3 points in 68 games last season and has hit 40 percent of his three-point attempts in eight NBA seasons.