The decision comes just a week after the BBC received planning permission for the scheme from Westminster council.
Project sources say the contest is hotting up between Schal and Bovis. One said: "There are a lot of high-level meetings right now."
Another insider suggested that a decision could be made on Monday. The insider said: "The BBC wants to keep on programme with the project. It wants to start letting out some of the specialist packages pretty soon."
The five bidders shortlisted in March were asked to put forward a bid for a construction management role and another for a guaranteed maximum price. The maximum price quote will be used if the job is taken over by the BBC's property partner, Land Securities Trillium. The firm took over the BBC's London estate, except for Broadcasting House, last year.
It is understood that talks are in progress between Land Securities and the BBC. The developer is already working with Bovis on the BBC's £200m studio complex at White City in west London.
Portland Place will be home to BBC Radio and Music, BBC World Service and BBC News. It will include a public arcade linking Portland Place and Hallam Street.