Griffey became the sixth player in Major League history to reach the 600-homer plateau when he sent a blast into the right-field seats in the first inning against the Marlins on Monday at Dolphin Stadium.
The man who held onto the ball after the scramble, wearing a giveaway Sergio Mitre jersey, would not identify his name and refused to speak to the media.
Marlins president David Samson spoke on his behalf and said his name is Joe, is in his mid-40s and has been a season-ticket holder since 1993. Samson said he has met him at a few of the Marlins' events.
Joe has not decided what he will do with the ball yet. After he had possession of the ball, he was escorted to the ground floor of Dolphin Stadium and put in one of the executive offices. Samson gave him his personal phone number and told him to sleep on his decision.
"He's a Marlins fan, he recognizes the importance of the ball, and he recognizes the importance that it would have to Ken, who he called a class act and a first-rate player," Samson said. "He's going to call me [on Tuesday]. We're going to talk about the ball and what the best course of action is for it."
Samson said Joe caught the ball on the fly, pointed to the floor to make others think the ball was there and fled the scene.
But 25-year-old Justin Kimball, who lives in Miami, said he initially caught the ball, held on to it for at least two seconds and then had it snatched from his hands before he could drop to the floor and cradle it.
"It corroborates it on video evidence that I caught the ball, and I have a bunch of people here saying I caught the ball," said Kimball, pointing to scratches on his leg and arm. "I reached, I caught it, I brought it back and the guy just ripped it out of my hands."
Kimball's lawyer, Ariel Saban, made it to the game and said his client indeed caught the ball first. He had no comment on whether they would take any legal action.
Several other witnesses in the right-field seats, including the usher, said they saw the same thing. Zach Johnsen, 25, was coming down the stairs when he saw the brawl for the ball.
"[Kimball] had it for at least a couple of seconds," Johnsen said.
In the Marlins' offices, Reds media relations director Rob Butcher and Reds clubhouse manager Rick Stowe told the fan that Griffey would like the ball.
The fan replied to them: "Before you go any further, because of my circumstances, I'm keeping it. I have plans for it."
"Different people do different things," Griffey said. "I'd like it. I can't control it. The guy has it. I'll worry about it later."
Umpires have been switching in specially marked balls since Griffey hit No. 599 on May 31. The 600 ball was properly authenticated by MLB officials in front of Butcher and Stowe and was considered legit.
With past homers, the Reds have offered fans Griffey autographs, bats and a chance to meet him in exchange for the ball.
Samson said Joe will not meet with Griffey after the game and will go home instead.
"We're going to use the ball to cover cost overruns," Samson joked. "He had a hunch that he may pull a ball tonight, and he was right. But he is, above all, a fan of the Marlins, and his focus is on the Marlins, and he was looking forward to speaking to me tomorrow about everything."
Alden Gonzalez is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.