The firm is battling it out with Lend Lease for the construction management job after other rivals Mace and a Skanska/Tishman joint venture dropped out of the running.
A source close to the project said: “The word is McAlpine is close to a deal, but nobody is resting on their laurels.
“I wouldn’t be surprised if Skanska Tishman make another pitch. They have a strong relationship with Bloomberg in the US.”
The building has been designed by Foster and Partners and will be up to 500,000 sq ft, large enough to house modern media services, such as studios for its TV facilities.
Bloomberg’s new headquarters will be built on the site of the former Bucklersbury House, between Bank, Cannon Street and Mansion House.
It’s owners Legal and General had previously worked up a detailed scheme for the site nicknamed “Darth Vader’s Helmet”, in partnership with Spain’s largest developer Metrovacesa, but this fell by the wayside when the firm pulled out last year.