The museum wants to sandwich a futuristic building, The Spiral, between its Victorian galleries in Kensington, but the council's planning officials recommended yesterday that permission be refused at next Monday's committee meeting. They claim the building, designed by the avant-garde architect Daniel Libeskind, is too large for the site.
The extension is part of a pounds 75m project to provide extra gallery space at the museum, but it has attracted support and opposition in equal measure ever since the design was unveiled two years ago.
The seven-storey building was designed to sit between the museum's Henry Cole Wing and the Aston Webb site in south Kensington. The architect has described it as "a geometric spiral", faced with tiles which, higher up, would give way to glass panels. Lifts would scale the building's exterior and it would house galleries and an education centre.
Sir Jocelyn Stevens, the chairman of English Heritage, said: "It will be a tragedy and a wasted opportunity if it is turned down. A building can change the whole perception of a town. The Sydney Opera House changed Australia and this building is as important."
Gwyn Miles, head of major projects at the V&A, said she was very disappointed with the council's recommendation but was confident that the museum would win any appeal. "This is a building that the V&A needs for its contemporary art, craft and design. But it will also be a centre for London and the country," she said.
But many local residents are opposed to the extension. A spokesman for the Chelsea Society said the building was deliberately designed to shock and "Exhibition Road is not the Bilbao waterfront".
Carol Seymour-Newton, of the Knightsbridge Association, said the development lacked "dignity and grace".
A spokesman for Kensington and Chelsea Council said it had received 22 letters of objection and 20 of support.