It plans to create a major educational hub scheme at the historic building in the Quartermile development area to be called the Edinburgh Futures Institute.
Sir Robert McAlpine has completed the asbestos strip and enabling works under a £3.6m deal.
Kier and Graham Construction are presently also working for the University on major projects.
The University is now seeking to prequalify six contractors to bid for the bulk refurbishment and new build works at the site
The University plans to restore and extend the existing buildings to form a wide range of teaching and learning spaces under plans drawn up by architect Bennetts Associates.
The proposals provide around 230,000 sq ft of floor space, of which 65,000 sq ft is new construction.
New four-storey “infill buildings” will be constructed in the two courtyards between the east and west hospital wards and will provide space for lecture halls and teaching spaces.
Main construction is expected to start next summer, opening to student in mid-2021.
