No one was injured in the fire at the building on The Strand, which broke out around midday on Saturday.
Somerset House is home to a range of creative businesses and public initiatives, as well as The Courtauld Gallery, which escaped damage and will reopen on Sunday. The gallery's collection includes works by Impressionist and PostImpressionist greats such as Van Gogh and Monet.
The building itself will remain closed until further notice as an investigation into the fire is conducted.
London Fire Brigade said 125 firefighters and 20 engines helped to tackle the blaze, which was brought under control just before 7pm.
Somerset House Trust director Jonathan Reekie said the fire broke out in the west wing, which is mainly comprised of offices and "back-of-house" facilities. He said no artworks were in the area.
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