The walkout, from February 3 to 7, will cause "serious disruption", said the Public and Commercial Services Union (PCS).
Some 250 workers are involved in the dispute. A recent ballot saw workers, on a 62 percent turnout, vote by 94 percent in favour of taking industrial action.
The museum says it wants to appoint an external partner to manage a new staff roster for extended opening hours.
The PCS claims the changes would involve the privatisation of most staff.
"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."
Situated on Trafalgar Square, the National Gallery is one of the British capital's more important tourist attractions and one of the world's most visited art museums. It is free to enter.
National Gallery director Nicholas Penny 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.
PCS National Gallery staff went on strike in October in a bid for better pay. Some sections of the gallery shut down altogether.
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