The £1bn+ contract is one of the largest building awards of the year.
The Enquirer understands Multiplex has beaten off rival bidder Mace to land the prized first phase of the vast town centre regeneration project.
Sources close to the scheme said developer Delancey is set to confirm the award imminently.
Work includes demolition of the iconic shopping centre and Hannibal House office block above and its replacement with new retail outlets and residential towers of up to 34 storeys alongside a new building for the London College of Communication.
Delancey has used Mace previously as construction advisor for the huge phased scheme which has a development value of £4bn.