The firm - which won a RIBA conservation award this year for its work on Chandos House in central London - has been taken on to renovate the Grade II-listed 1733 house on Saville Row.
It will be converted to become the new headquarters of property investment firm CUBE, following an extensive survey by Met Surveys.
Basevi, whose most important building is considered by many to be the Italianate-style Fitzwilliam Museum in Cambridge, is also well known for designing classical London terraces including Belgrave Square.
He is renowned as one of John Soane's most famous - and most successful - students. Examples of Basevi's work can be found at the Sir John Soane Museum.