The pub adjoins Bramwell House on the Rockingham Estate.
Blazeland Ltd – part of Acorn Property Group – has applied to Southwark Council for permission to demolish The Rising Sun and build a six-storey block of nine flats, with a commercial unit on the ground floor.
The scheme has been designed by Southwark-based architects Fereday Pollard.
In their response to Blazeland's pre-application enquiry last autumn, Southwark planners warned that the developer would have to work harder to prove that the pub was no longer viable.
"No marketing evidence has been submitted to justify the loss of the pub, which is likely to hold social value for the residents of the estate," wrote case officer Glenn Ruane.
In terms of design, Mr Ruane said: "Whilst the existing public house building on site is considered to be an attractive building, its demolition and redevelopment of the site is considered acceptable in principle subject to the development of a high quality scheme in its place."
The scheme follows the redevelopment of the Royal Standard pub by the same developer.
The former William IV pub – most recently used by an immersive theatre company – is also in line for redevelopment.
For more on the scheme see planning application 20/AP/1374