"Sir Charles" made the comments about Green on Tuesday, saying he just wants "somebody to punch him in the face." He was watching a clip of Green and New Orleans Pelicans guard Rajon Rondo talking trash.
"I want to punch his [expletive] in the face, I do," Barkley added, while serving as an analyst on NBA on TNT.
Barkley made the comments at halftime of the game, which Golden State won 121-116 to go up 2-0 in the Western Conference semifinals.
After seeing his comments, Green's mother spoke up on Twitter.
Dray dropped the mic on Charles. pic.twitter.com/KRCVzFdpWx— Warriors on NBCS (@NBCSWarriors) May 2, 2018
"Charles Barkley, I wish you would ... you think Shaq had you running! $ManGone," she tweeted.
"He would not have done nothing but run like he did when he started the fight between Dr. J and Bird," she added.
"Dray ain't scared ... please believe that! Win or lose, he ain't running like Charles did," she wrote.
Green was asked about Barkley's comments when he met with the media, following the Warriors' Game 2 victory.
"A lot of guys talk on TV and stand behind the microphone on a TV screen, but the fact of the matter is if you feel that strongly about something ... he has seen me a million times," Green said. "If he feels that strongly about it then punch me in my face when you see me."
"If you're not gonna punch me in my face when you see me, then shut up. It's no different than somebody sitting behind a computer screen tweeting 'I'll knock you out!' And you've never seem them in real life. Well he has seen a bunch of times and he'll see me again this year. Punch me in the face when you see me and if not, no one cares what you would have done. You're old and it is what it is. So if you ain't gonna punch me when you see me, then stop talking about it, period."
Green scored 20 points and had 12 assists and nine rebounds in the Warriors' win. The Warriors and Pelicans resume their series at 8 p.m. Friday at the Smoothie King Center in New Orleans, La.