The new building would neighbour the Tate's prominent site on the South Bank of the Thames.
The Tate confirmed late last week that it had been in discussions with the museum, currently located in Shad Thames.
A Tate spokesperson said: 'Over the last six months, Tate has been in discussion with a number of potential partners in connection with the project to develop Tate Modern and the surrounding cultural quarter.
'As part of this process, there have been meetings with the chairman and director of the Design Museum to discuss a possible relationship. This dialogue is ongoing.'
The announcement of the museum's move coincided with the high-profile departure of its former director Alice Rawsthorn at the end of January.