Six locations are being looked at for the new building, which would be double the present size. Riverside positions in particular are under consideration, with one at Blackfriars Bridge getting close scrutiny.
Consultants Allsop and Stormer are due to present their first and second choice sites to ICA management next month. An international competition to find architects will be launched in October.
Mik Flood, director of the ICA, said the listed Regency terrace designed by John Nash was no longer large enough to house the proliferation of art exhibitions, drama, live music, cinema events and workshops.
'Although the Mall building has served us well it can't sustain the sort of artistic growth we've seen over the past
decade. If the ICA is to maintain its key position as one of the leading arts centres of Europe it needs to be able to host works which utilise new changes in technology, from interactive installations to video and multimedia shows.
'We want to put something into London of world stature. The design has to be technically advanced, highly flexible, and really exciting, and would look towards the next century.
Mr Flood hoped the bulk of the pounds 10m-pounds 15m cost would be met by the Millennium Fund, which is being raised through the national lottery and administered by the Department of National Heritage.
Other sources of money could be the Arts Council, local authorities, and the British Film Institute. The complex could be ready within four years, but until then the ICA will stay where it is.
Mr Flood is confident users will welcome the relocation as reflecting the spirit of the ICA. 'When we moved from Dover Street in Soho 25 years ago some members sent their cards back and accused the Institute of selling out. A few people may not like change, but the new site will be high-profile. It will be a new cultural centre for Europe of the same artistic scope as the South Bank.
Between 1992 and 1993 the ICA held more than 2,200 events which attracted well over 200,000 visitors. It recorded an operating surplus of pounds 42,000. Its notoriously hip bar is a favourite haunt for arts lovers wanting to hang out and be seen looking cool.
The institute was founded in 1948 by poet and critic Sir Herbert Read, and the surrealist painter Sir Roland Penrose. It is Britain's most influential international centre for contemporary arts and a must for many overseas visitors - Richard Gere was spotted there on Wednesday night. The singer Madonna is vice-president.
Incongruously for a building renowned for hosting radical events, its location on the Mall - sandwiched by Buckingham Palace and the Ministry of Defence - places it at the heart of the Establishment.
The complex has long attracted criticism from traditionalists and politicians. In December 1983 its exhibition of explicit homoerotic photographs by the late Robert Mapplethorpe created controversy.
Last year its deliberately confrontational Bad Girls show featured a piece by Piss Flowers artist Helen Chadwick, and paintings of sexual violence and child abuse by American Sue Williams. It has also hosted fetish and piercing conferences.