Sources close to the project said that Kier is set to take the £35m refurbishment job, which is due to start later in the year.
The 1970s Arup Building is home to the Museum of Zoology, the Babbage lecture theatre and the Department of Materials Science and Metallurgy.
The department plans to relocate this year and the university is taking the opportunity to refurbish and upgrade the whole of the 150,000 sq ft Arup Building.
The building – on the New Museums site – was designed by Arup Associates in the 1960s and will reopen in its new form in 2015.