Mace is understood to be the front-runner to take the £57m project to build two new medical facilities at the University’s Old Road Campus, by the Churchill Hospital in Headington.
If confirmed successful, the firm will build a £20m Nuffield Department of Clinical Medicine and a £37m Institute for Rheumatology.
Mace, which is bidding the job alongside several other rivals including Laing O’Rourke, is understood to have had its Bills called in first.
Three buildings need to be demolished to make way for the new buildings.
Construction work is scheduled to begin in October for a 2013 opening.