Video was posted online of the incident, which came after the Suns' 130-88 loss in Los Angeles, showing Jackson making the gesture and shouting an apparent profanity at the fan.
Suns general manager Ryan McDonough told reporters that the NBA has reached out to Jackson, who said he has seen the video.
"That's what most people thought I was [doing] but I actually wasn't making a gun," Jackson said, according to azcentral.com. "I kind of wanted to put up the middle finger to him but I didn't do that because I felt like I was really being watched, so I kind of halfway did it."
According to Jackson, the fan had been heckling him throughout Saturday's game. He said he needs to learn to not react to fans in that manner.
"He called me a few names as I was coming out of the game so that's what really set me off," Jackson said, according to azcentral.com. "But it had been going on from the same guy the entire game. Fans can come and sit so close and say whatever they want to say and us as players can't react in any type of way. It's something I'm still learning."