The Athletic's Shams Charania claims that Green and Poole needed to be separated after a heated interaction escalated during practice on Wednesday.
The players reportedly came chest to chest and pushed and shoved one another, before Green escalated the altercation.
The incident, which occurred on the second day of Golden State's training camp, forced the Warriors to stop practice and exit for the locker room to defuse the situation.
Four-time All-Star Green, who had 7.3 rebounds and 7.0 assists per game last season, is known for his physical approach to the game.
The Warriors are preparing for their title defence in 2022-23, after beating the Boston Celtics 4-2 in last season's NBA Finals.
Poole enjoyed a breakout season in Golden State's triumphant 2021-22 campaign, averaging a career-best 18.5 points, 3.4 rebounds and 4.0 assists, and is in line for a big contract extension.