The three in the race for the Wynyard Hospital are: Brookfield Multiplex Europe, Iridium Concesiones de Infraestructuras and Laing O’Rourke.
Bosses at the trust aim to fund the superhospital from pension fund loans, the first time this route has been used in the UK.
A funding competition will run in parallel to identify the best terms available to fund the scheme.
Kevin Oxley, lead director for the new hospital, said: “We had many people at the bidder open day in October and 10 organisations submitted pre-qualification questionnaires to be shortlisted for the next stage.
“We had a number of strong submissions of which Brookfield, Iridium and Laing O’Rourke offered the best technical and financial standing.
“These bidders have been invited to work with our doctors, nurses, other health professionals and support staff on the design and delivery of the scheme.”
Trust board chairman Paul Garvin described the funding approach as similar to taking out a mortgage.
He said: “We borrow the money, we build the building, and we pay it back year on year.”
A PFI project could have cost £660m over the 30-year life of the scheme.
The bid process is expected to take 18 months. After that the Trust board will take a decision on whether to borrow the money, award contracts and begin the build.
The project is a smaller-scale version of the original £460m plan, which was scrapped, along with 12 others nationwide, in June 2010 as part of measures by the coalition government to save £2bn.