A team led by IT firm Logica with contractors Costain and Skanska is vying for the high-tech scheme with an Amey/Mace consortium. A decision is due to be made on the winning team in April.
The two teams were shortlisted last month following the Met Office's decision to move its headquarters from its present site in Bracknell to the West Country.
Logica's team also includes architect Broadway Malyan and security outfit Group Four, while Amey/Mace are working with architect Alsop and St枚rmer on its proposals.
The scheme, funded by the Met Office itself, will be a mixture of design and build for the construction of the headquarters and a 15-year facilities management provision for the upkeep of the building.
Procurement manager Rob Varley said the Met Office had considered the PFI route but had decided that funding the scheme itself would be better value.
He said: "I think PFI is quite good for production line projects such as prisons but with our scheme, which is unique, it would have taken a long time to get off the ground through that route. We are very comfortable with the route we are taking."
The Met Office announced last month that it had chosen the site, just west of the Exeter Business Park, from a shortlist of locations that included Reading, Norwich and another site on the outskirts of Bracknell.
The building will be a major global hub handling an unimaginable network of data
Rob Varley, Met Office
The Met Office first announced plans to move to new headquarters in June when it asked bidders to put forward proposals for the scheme.
One team insider said the decision on the eventual location of the project came as a relief.
He said: "It has been difficult working on a proposal when you don't know where the site is."
The Met Office claims the scheme will involve the largest IT relocation project in Europe.
The headquarters will house Met Office super computers, the most powerful in the world outside of North America, and therefore need complex building services to control the atmosphere within the building.
Varley said: "It will be a major global hub which will handle an unimaginable network of data from satellites, radars and observation posts."
The decision to move the headquarters was made because the Met had outgrown its present accommodation.
