The centre, due for completion in 2006, will house two recording studios for the BBC Symphony Orchestra and other ensembles.
The competition is being chaired by John Smith, the BBC's director of finance, property and business affairs. He said: "We are looking for an iconic building and are not afraid to champion a potentially controversial design."
The music centre is part of the BBC's media village complex in White City, designed by Allies and Morrison, which will include recording studios and offices.
The music centre is the latest in a series of projects for the BBC. Others include the £250m redevelopment of Broadcasting House, which was awarded to Bovis Lend Lease in January. Last month, the BBC also commissioned architects Hodder Associates to design a regional headquarters in Manchester.