The University is planning to build a £20m Nuffield Department of Clinical Medicine and a £37m Institute for Rheumatology.
Three buildings will be demolished in Headington to make way for the new buildings at the Old Road Campus.
Mace, which is bidding the job alongside several other rivals including Laing O’Rourke, is understood to have had its Bills called in first.
A full planning application was submitted to Oxford City Council three months ago and construction work is scheduled to begin in October for a 2013 opening.