Developer Countryside and Swan Housing are up against rival consortium London Quadrant and Telford Homes for the scheme, which has been dubbed Blackwall Reach.
London council Tower Hamlets is expected to name the successful bidders for the scheme to demolish the 215 apartment estates in December.
Plans to develop the estate ran into controversy last year as architects lobbied to save the 1970s estate, claiming it is a seminal building of its time.
Among the campaigners was Lord Rogers who said the estate designed by architects Alison and Peter Smithson should be saved.
During consultations about the replacement housing scheme, more than 75% of residents said they would like to see Robin Hood Gardens knocked down and replaced.