A yellow weather warning from the Met Office is in place for Sunday, with gusts of up to 52mph expected throughout the day in the Harrogate district.
A spokesperson for the National Trust said: "We have taken the decision to close the estate (including the Studley tea-room) on Sunday 9 February due to the forecast high winds. This is for the safety of our staff and visitors. We also ask that visitors avoid walking in the Studley deer park for their personal safety.
"The entire Fountains and Studley estates will be closed, including car parks. Obviously we can’t completely close off access to the deer park/seven bridges, but are asking people to avoid visiting these areas also for their personal safety."
Rain is also forecast across the district on Sunday, with expectations that it will be heavy between 6am and 3pm.