Southwark Council leader Peter John says the the fate of items in two galleries at the Cuming museum is 'unknown at present'.
Firefighters tackling the fire at Walworth Town Hall are currently damping down.
On twitter, Councillor John paid tribute to the London Fire Brigade and to Southwark Council for their efforts in trying to save the museum's collections.
Amazing work by LFB and Southwark Council museum staff to remove Cuming Museum artefacts from Walworth fire today.
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She met with the families of children who must stay near the hospital.
Six people, including three children, died in the fire at Lakanal House in Camberwell in 2009.
PCs Gavin Bateman and Tony Stephenson spent 30 minutes drinking tea.