A Kensington Palace statement said they were prompted to take action after the man was spotted near the 14-month-old and his nanny in a central London park.
The couple are not taking legal action at this stage, but may consider it in the future, the statement said.
Lawyers for the photographer, Niraj Tanna, said he had done nothing wrong and would continue with his work.
The statement from the palace said there was "reason to suspect that the individual may have been placing Prince George under surveillance and monitoring his daily routines for a period of time."
BBC royal correspondent Nicholas Witchell said palace officials believed the photographer's actions amounted to "stalking" and that lawyers were seeking an assurance he would desist.
"If they do not receive these assurances that this behaviour will stop, they will then consider options of legal action," he said.
Our correspondent added that 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 behaviour.
Not for the first time, this is Prince William attempting to define what constitutes his family's private life as he struggles - in the face of intense global media interest - to carve out as near as normal an existence as he can for his toddler son, who will one day be king.
In doing so, he's picked a very public fight with a photographer, Niraj Tanna, who is keen and determined to stand his ground.
Mr Tanna's defence is that he's entitled to take pictures in a public park and he hasn't followed, harassed or intimidated either George or his nanny.
William's next step could be court action.
In recent years, royals and court cases have proved to be uncomfortable bedfellows.
You can read more from Peter Hunt on his BBC royal and diplomatic correspondent page.
However, lawyers for the photographer Niraj Tanna have issued a seven-page letter contesting the claims.
The letter says Mr Tanna strongly objects to what he calls "the groundless allegations".
It says it is "wholly without foundation" to accuse him of following or spying on them.
It goes on to say that press photographers are "fully entitled" to take images in public places such as parks, and that any legal action will be "vigorously contested".
It adds: "He will continue to undertake his work with the concerns of the Prince's parents very much in mind."
Since his birth in July 2013, the royal couple have posed for a number of official photographs with Prince George.
He has also been pictured accompanying his parents on state visits and with the palace's consent at other events.
The statement from the Kensington Palace said: "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."
It added: "No parent would tolerate the suspicion of someone pursuing and harassing their child and carer."
Maria Teresa Turrion Borrallo was named as the prince's nanny in March when the palace appealed for her privacy to be respected.
The Independent Press Standards Organisation (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.
Before the royal couple married, the Middleton family raised privacy concerns about alleged harassment by press agency photographers.
And in 2012, lawyers for the pair also took action against French gossip magazine Closer for publishing topless photos of the duchess.
Lawyers for the Duke and Duchess of Cambridge have asked a photographer to "cease harassing" Prince George.