Sources close to the project said the firm looked set to seal a deal ahead of rival bidders ISG and Sisk.
The building – which once housed prisoners including Casanova, Oscar Wilde and the Kray twins – will be partially preserved and transformed into a 99 bed hotel.
Of the total hotel rooms, four bedrooms will be created from 10 of the existing police cells, located on the first floor of the Martlett Court building.
The main court rooms will be restored and turned into a restaurant.
Work is expected to start before the end of the summer, subject to full planning approval, and take around 21 months to complete.
Part of the magistrates court and police building will become a living museum.