The PCS union is running an indicative ballot for strikes among warders at the National Gallery in London.
Workers oppose cuts to staff numbers. They believe new lower staffing levels made it possible for a man to vandalise two Poussin masterpieces last July.
Strikes could begin during a blockbuster exhibition of rare Leonardo Da Vinci paintings which opens in November.
If warders do walk out it will force the evacuation of the gallery.