According to the San Diego Union-Tribune, Greene suffered the injury during a 7-3 loss to the Diamondbacks when he hit or punched a storage chest outside the home dugout during the game.
Greene apparently suffered the fracture in the seventh inning when he struck out for the 100th time this season.
San Diego general manager Kevin Towers said Thursday that Greene would be placed on the disabled list and that he could miss six to eight weeks. Greene was to have an MRI exam on Thursday.
Towers was asked if Greene could face any disciplinary action for the way he injured his hand.
"I need to get more of the facts of what happened, the nature of the injury before we decide what to do moving forward," Towers said.
Greene is hitting .213 with 10 home runs and 35 RBIs in 105 games, and is well behind his pace from a year ago when he hit .254 with 27 home runs and 97 RBIs.
The Padres will use Luis Rodriguez and possibly Edgar Gonzalez at shortstop while Greene is gone. Tadahito Iguchi, who is on a Minor League rehabilitation assignment as he continues to work his way back from a separated right shoulder, will be reinstated from the disabled list prior to Friday's game against the Giants.
Gonzalez has capably filled in at second base for Iguchi, though Rodriguez is considered to be a better defensive option at shortstop.
Corey Brock is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.