A third bid from Laing/Hyder did not make it to the final stage for the 800-bed hospital. Havering Hospitals NHS Trust project director Duane Passman said the decision to reject this bid was not related to the sale of Laing's construction arm.
A preferred bidder will be chosen by August with financial close expected in December.
Catalyst Healthcare is also bidding for the redevelopment of Walsgrave Hospital in Coventry. A shortlist for the £200m scheme will be announced next Friday.