The run at the Duchess Theatre will now run from 18 February to 8 March.
Marti Webb reprises her original role in the show, having created it over 30 years ago. The album Tell Me on a Sunday reached number one in the charts and the single "Take That Look Off Your Face" reached number two.
Tell Me On a Sunday sold out last month at St James Theatre. The West End transfer is produced by Barry Satchwell Smith, Robert Mackintosh, St James Theatre Productions Ltd and Nimax Theatres.
WhatsOnStage's Mark Valencia said of the production: "The extended version of Tell Me on a Sunday that Lloyd Webber and Black prepared for Denise Van Outen a decade ago was updated and coarsened in a way that diluted the original show's simplicity and diminished its emotional impact, so it is a special treat see and hear its dedicatee restore the factory settings.
"I neither know nor care Marti Webb's current age; in the role she created three and a half decades ago she's still the naïve young woman who stands alone before us, learning life's lessons the hard way, abetted by nothing but a handful of props and Simon Lee's immaculate seven-piece band."