An X-ray and MRI revealed that Green has a nondisplaced fracture of his second metacarpal.
Green has played in 38 games this season, including 28 as a starter, and is averaging 7.4 points and 3.0 rebounds. He averaged a career-high 10.5 points last season.
Spurs guard Manu Ginobili has missed the past few games with hamstring tightness, but he is expected to return soon.
Oklahoma City Thunder small forward Kevin Durant did not practice Monday because of an injury to his left wrist.
The injury is to Durant's non-shooting wrist and is not believed to be anything other than soreness.
Coach Scott Brooks would not say if Durant would play Tuesday night against the Memphis Grizzlies.
Australian teenage sensation Dante Exum is likely to enter the 2014 NBA draft.
Exum met with eight major player representation agencies and gave them a strong impression he plans to enter the June draft, according to the report.
"I would be shocked if he doesn't enter the draft," one agent who recently met with Exum's family told Yahoo Sports.
Because the 6-6 Exum is an international player who will turn 19 during the 2014 calendar year, he is eligible to enter this year's NBA Draft without having to spend a year in college. He is a likely lottery pick.
The Minnesota Timberwolves recalled rookie guard/forward Shabazz Muhammad from the Iowa Energy of the NBA Development League.
The Detroit Pistons recalled rookie guard Peyton Siva and rookie forward Tony Mitchell from the Fort Wayne Mad Ants of the NBA Development League.
The Sacramento Kings assigned rookie guard Ray McCallum to the Reno Bighorns of the NBA Development League.
The Toronto Raptors recalled rookie guard Dwight Buycks from the Bakersfield Jam of the NBA Development League.