The bridge at the park in the Dukeries near Worksop built in the 1760s can no longer carry pedestrians or vehicles.
The extent of the damage is currently unknown but police were called to reports of the incident at 8am today (Saturday, March 3).
A spokesman for the National Trust who own the park said: "We regretfully inform visitors that, due to an act of intentional vandalism and resulting significant structural damage, Clumber Bridge will be closed to pedestrians and vehicles for the foreseeable future.
"The south side of the park can still be accessed 'clockwise' via the lakeside path and through Hardwick Village, however, please note that the usual 'lakeside loop' cannot be undertaken and upon reaching the bridge there is no way to cross the river."
A spokesman from Nottinghamshire Police said: "We were called at 8am to reports of damaged caused to a bridge and an abandoned vehicle in Clumber Park.
"Officers attended and an investigation is in its very early stages.
"This will affect visitors to the park and will undoubtedly have an impact on National Trust members and the local community."