The 180,000 sq ft complex, which will be built on a sports pitch on the western edge of the Broomhill campus, has just been given the planning green light.
Sources told the Enquirer that Kier is favourite to take the project after fending off rival bids from BAM and Graham Construction.
Construction work on the Social Sciences building will start on site in early 2018, with an expected completion date of spring 2020.
The two-year project is expected to cost around £50m to build uniting three departments – economics, politics and sociology – as a ‘hub’ under one roof.
Currently the Social Sciences estate is spread across 19 different buildings across campus.
Last month, plans were approved for two new artificial pitches for rugby, football and hockey at the university’s playing fields site at Norton, to avoid losing sports provision.
Plans were also approved this week for a new sports centre on Northumberland Road, which will include a new pool and greater gym capacity.