Fire crews were called after smoke was seen coming out of an inspection cover on Kingsway, in Holborn. The fire service said it was waiting for the power to be turned off before it could put out the blaze. Crews are working to prevent the fire spreading to buildings in the area.
Shows at the Lyceum, Duchess and Aldwych theatres have been cancelled, London Theatre Direct said.
Earlier, judges, lawyers and other staff at The Royal Courts of Justice in The Strand, as well as buildings along Rose Street and Garrick Street and the market building in Covent Garden were evacuated.
The fire has caused 10 bus routes to be diverted, according to Transport for London (TfL), and Holborn station is closed.
Kingsway is closed between Holborn and Aldwych, the Strand Underpass and Waterloo Bridge northbound, causing serious delays according to the BBC.