English Partnerships launched the scheme last September. Profits of the scheme, which aims to deliver more than 15,000 homes in the next five years, will be split between EP and the housebuilder.
Other companies believed to be bidding include housebuilders Lovells and Genesis, developer Quintain and innovation consultant Mtech.
EP has acquired the first four sites for the housing in Holloway Road and Islington in north London, and Steedman Street and Croydon in south London. It will also use a site in East India Dock in London Dockland that it already owns.
An Official Journal notice was issued in April for a private-sector development partner. The partner should be in place by the end of this year and development could begin next year.