The Cavs are looking for two-way wing players like Green as they look to retool their roster to better compete with the NBA champion Golden State Warriors.
The Spurs have been investigating the market for Green as they look to open salary-cap space ahead of the start of free agency. But the asking price has been a little too high in early talks with the Cavs, sources said.
Green has two years and $20 million left on his contract. He averaged 7.3 points and shot 38 percent on 3-pointers in 68 games for the Spurs last season. He is also a strong defender.
The Cavs currently don't have a general manager after the team and David Griffin parted ways earlier this week. Cleveland has had multiple meetings with Chauncey Billups about running the front office.
In the meantime, Cavs executives Koby Altman and Brock Aller have been working the phones in recent days, with owner Dan Gilbert being active in personnel decisions.