Frank Haslam Milan, Henry Boot, Herbert T Forrest, Kier and Lovell are all thought to have had an interview with the Ashfield Homes ALMO last week.
It is not yet known when an award for the job will be made, but work is set to start in November.
The works involve the replacement of kitchens, bathrooms and central heating, as well as electrical rewiring on around 7,000 council-owned properties within the borough.
The contract is set to last until March 2012, with an option to extend for a further 36 months at 12-monthly intervals.
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