Among those detained is Bellarmine Mugabe, who is believed to be in his twenties.
A 23-year-old male, possibly a gardener or security personnel, was reportedly shot during the incident.
The injured man has been transported to a medical facility and is reported to be in critical condition.
Police spokesperson Col Dimakatso Nevhuhulwi shared with the BBC that investigators are still unraveling the details of the event.
Investigators are currently present at the Hyde Park site where the shooting reportedly occurred, searching for a weapon.
While police have not publicly identified Bellarmine Mugabe as a suspect, onlookers at the scene reported seeing him in handcuffs.
His name has become prominent in local news reports since the incident.
Earlier, an individual identifying himself as “Batista” and claiming to be associated with the family’s private security team informed SABC that he was called to the property for assistance.
He conveyed his discontent after being denied entry to the Hyde Park home.
Bellarmine Mugabe is the youngest offspring of Robert Mugabe and his second wife, Grace Mugabe.
The former president of Zimbabwe governed for 37 years before resigning in 2017. He passed away in 2019 at the age of 95.
The investigation is ongoing as authorities gather more details about the circumstances surrounding the shooting.
