A Kensington Palace statement said they were prompted to take action after the man was spotted near 14-month-old Prince George in a central London park.
The statement said: “No parent would tolerate the suspicion of someone pursuing and harassing their child and carer.”
Prince William and Kate Middleton are not taking legal action at this stage.
The man had been warned off last week by royal protection officers and that he had also been spoken to “over a number of years” about his behavior.
The Kensington Palace statement added: “There is reason to suspect that the individual may have been placing Prince George under surveillance and monitoring his daily routines for a period of time.
“The duke and duchess understand the particular public role that Prince George will one day inherit but while he is young, he must be permitted to lead as ordinary a life as possible.”
Maria Teresa Turrion Borrallo was named as the prince’s nanny in March when the palace appealed for her privacy to be respected.
Since his birth in July 2013, Prince William and Kate Middleton have posed for a number of official photographs with Prince George.
Prince George has also been pictured accompanying his parents on state visits and with the palace’s consent at other events.
The Independent Press Standards Organization (IPSO) editors’ code of practice states that young children “must not be interviewed or photographed on issues involving their own or another child’s welfare unless a custodial parent or similarly responsible adult consents”.
IPSO replaced the defunct Press Complaints Commission last month.
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