Construction information specialist Glenigan has Laing O’Rourke and Lendlease listed as the two bidders on the shortlist to become management contractor.
The council was originally hoping to shortlist up to five firms when the tender race started in February.
The winning firm will be responsible for managing the project and procuring around 80 subcontracts ranging from £10,000 – £40m.
Construction costs on the project are estimated at £190m.
Works will include restoring the Great Hall and its Ford Madox Brown murals, as well as external repairs to the roof, stonework and drains.
The council said ‘significant work’ is needed to address the electrics, heating, ventilation and lifts in the building.
The client project team includes Mace who were appointed late last year as project manager and architect Purcell, which has worked up designs to refurbish the Grade I-listed landmark building.