The successful contractor will not be named until planning permission has been granted for the scheme, which has faced passionate opposition from residents and local businesses. Construction of the eight-storey leisure, retail and office complex is expected to take three years.
A special meeting of Tower Hamlets' council planning committee was held on Wednesday night to consider the Foster and Partners-designed scheme, which is supported by the Corporation of London.
Hammerson, the project's developer, has drawn up eight options for the site, near Liverpool Street Station, because it so sensitive. It is believed that the council will consider one or two of these options.
A contractor will not be named until the scheme, which has faced passionate opposition, gets planning permission
A project source said a swift decision would be made on the contractor if planning permission was granted. The source said: "We'll make our minds up quite quickly if the decision goes our way."
The scheme has been opposed by the Spitalfields Market Under Threat campaign, led by market stallholders who claim that the redevelopment will destroy the market's special character.
The campaigners are hoping that the redevelopment goes the same way as a £70m cinema complex proposed for Crystal Palace in south London. This was abandoned after a five-year struggle over planning permission.