Irizarry recorded the encounter on her phone, saying she felt threatened, and posted the video to Facebook.
"After the incident, we immediately launched an investigation pursuant to our personnel policies into the response of our officer," it said, in a series of posts on Twitter.
"The investigation is ongoing and the officer involved has been assigned to desk duty pending the outcome."
Timothy G. Trybus, the intoxicated individual involved in the incident, was arrested and charged with assault and disorderly conduct, according to a police report.
CNN has attempted to reach Trybus for comment.
"All people are welcome in the Forest Preserves of Cook County and no one should feel unsafe while visiting our preserves," the agency said.
On Tuesday, park spokeswoman Stacina Stagner said investigators will interview the officer this week about the incident. She said the investigation would wrap up soon, but did not elaborate on a timeline.
In the footage, the man can be seen approaching Irizarry saying: "You should not be wearing that in the United States of America." He gets closer to her and asks "Are you a citizen? Are you a United States citizen?"
Irizarry can be heard saying that Puerto Rico is part of the United States and the man approaches her multiple times.
Irizarry asks a park police officer to help, saying, "I am renting this area and he's harassing me about the shirt that I'm wearing."
Later she says: "Officer, I feel entirely uncomfortable, can you remove ... please officer" as the officer is seen walking away from her.
Then she says: "Officer, I'm renting, I paid for a permit for this area. I do not feel comfortable with him here, is there anything you can do?"
The officer can then be seen talking to the man who gesticulates back and tells him to "shut the f*** up."
More police arrive and Irizarry says she still doesn't feel safe. The man resumes his abuse, saying: "You're not American, if you were American you wouldn't wear that. You know that right?"
A female officer asks to see his ID and can be heard telling him that he's intoxicated, to which he replies, "Well that's your judgment." She explains that Irizarry has a permit and she warns him that he could be arrested "for not being compliant."
"You don't come here harassing people," the officer continues. "People have just as much right to be here as you and when you're drunk, you don't belong here."
The female officer then speaks with Irizarry, who gives her version of the incident and the officer explains that they were called to the area after a report that a man was choking a woman.
Eventually the first officer on the scene takes notes of Irizarry's account of the incident and says that he was at the scene due to the separate incident, noting that she was not being attacked though acknowledging she felt threatened.
Irizarry can be heard explaining to the officer that the incident began when they had asked the group the man was in if they could move as they had a permit for the area. She said the group politely complied but her Puerto Rico shirt appeared to act as a trigger to the man.
Puerto Rico Gov. Ricardo Rossello was on CNN's Erin Burnett "OutFront" on Tuesday night discussing the incident. He said he was "shocked, appalled and disgusted" when he watched the video.
"This was an attack of an American citizen on another American citizen," he said. "Puerto Ricans have been part of the United States and we've been fighting wars with other fellow Americans. We are proud US citizens. People need to understand that."
CNN's Marlena Baldacci contributed to this report.