The application, to North Yorkshire Council, says the wooden fence is to “provide a simple visual and modest physical barrier to discourage incursion onto the residential property”.
It follows a protest by Greenpeace in August that saw activists demonstrate on the roof of the grade II-listed mansion.
Campaigners draped the property with oil-black fabric to protest against plans to grant more than 100 new licences for oil and gas extraction in the North Sea.
Five people were arrested following the protest, which took place while Mr Sunak and his family were on holiday in California.
The planning application, which is currently awaiting a decision, to erect a “a timber post and rail fence” measuring 165ft long and 4ft high was submitted on October 11.
The document states: “It is considered that the proposal gives rise to no perceivable harm in heritage terms or from a wider planning perspective and planning permission should be duly granted.”
The Mail on Sunday, which first reported the application, said Downing Street had confirmed that Mr Sunak would pay for the fence himself.