The possible move emerged at a press conference held yesterday (25 July) to unveil Herzog & de Meuron's plans for a new extension to the Tate building.
It is understood that the Design Museum - whose new director Deyan Sudjic officially takes up his role in October - will hold an architectural competition for the new building if it confirms the move.
However, all parties are keen to emphasise that the talks are still at an extremely early stage.
If the Design Museum was to relocate to Bankside, it would be a key element in Tate director Nicholas Serota's long-standing desire to turn the area into a new cultural quarter for London.
This artistic district could expand as far south as Elephant and Castle, with a variety of attractions - of varying scales - littered throughout the 'cultural park'.