About 250 workers are involved in the row, which sparked a 9-1 vote in favour of industrial action in a recent ballot.
Talks were held at the conciliation service Acas but ended without agreement.
PCS general secretary Mark Serwotka said: "This five-day walkout will put the National Gallery in the spotlight for all the wrong reasons.
"The sell-off plan is reckless and risks damaging the worldwide reputation of what is one of the UK`s greatest cultural assets, and we are determined to stop it."
Nicholas Penny, director of the National Gallery, said: "If the National Gallery is to continue to thrive as a public entity with reduced public money, change is essential. There is no option that allows everything to stay the same.
"The proposed changes are necessary to enable the National Gallery to increase income in the face of a reduced grant and increasing maintenance and running costs, and to enable it to pay all staff a minimum of the London living wage.
"The PCS union leadership opposed the change and, despite five months of dialogue, we were not able to meet an agreement."
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