The trendy Mayfair location is already home to other flagship stores including Jimmy Choo, Christian Louboutin and McQueen's diffusion line McQ.
The successful fashion designer is in talks about taking the lease of number 36, according to the Evening Standard, though a spokesperson for the star has declined to confirm the news, saying, "A store in London is under discussion."
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Fellow British-based brands Christopher Kane and Roksanda Ilincic are also said to have set their sights on the location for their boutiques.
The news comes just days after Victoria launched her autumn/winter 2013 collection for Victoria, Victoria Beckham on the e-commerce site she debuted earlier this year.
Back in February, the fashion designer revealed that this collection was aimed at the "intellectual woman".
"I feel like the Victoria, Victoria Beckham girl this season is an engaged intellectual with a sense of humour," explained Victoria.
"She can wear anything with aplomb yet she chooses the simplicity of a dress. She marches rather than walks, she drinks espresso, and she devours books and art and film. And she is subversively sexy without trying. I'd like to meet her…"
Talking about designing the collection, Victoria added, "I'm really excited about this collection, it's effortlessly cool and easy and we've developed a lot of textures – it looks like there's a lot of layers.
"The idea is that you don't have to think about it. It looks like two pieces – jumpers and skirts – but it's just one piece and you can just zip yourself up and off you go."
The new line reflects Victoria's personal style – simple yet elegant dresses, menswear-inspired tailored pieces and utilitarian, no-frills fashion.
The androgynous vibe is grown-up, thanks to monochrome silhouettes and a muted colour palette – featuring tones of burgundy, copper and oxblood.