The company has charged the architect with investigating how more residential accommodation could be squeezed into the Grade II-listed landmark.
The move is likely to provoke outrage among the Brunswick's current residents, some of whose views would be blocked by construction work, according to the architect.
Hodgkinson made sure that the Brunswick's main residential units were structurally strong enough to bear the load of another residential storey when he worked on the building in the 1960s.
These units sit either side of a central shopping street, currently the focus of Allied London's £20 million refurbishment work for the centre.
Levitt told the AJ: 'It's going to be a sensitive issue because the building is listed and the views of the existing residents are likely to be particularly trenchant.'
Allied London also has planning permission for a new 'eye-catching' building on the centre's eastern side in front of the Renoir cinema.
According to sources close to the developer, it is also investigating the possibility of building this out. This comes after a new Waitrose supermarket at the centre was topped out last Thursday.