The final decision will be taken on Monday following a recommendation that the firm builds the job, beating rivals Balfour Beatty, Lend Lease and Sir Robert McAlpine.
If approved, a tender report will be brought back to the council in due course seeking agreement for a fixed-price contract.
Designed by Japanese architect Kengo Kuma, V&A at Dundee is the first major design museum to be built outside of London.
It is also the anchor project for city’s ambitious waterfront development .
Council administration leader Councillor Ken Guild said: “This is a major milestone in the project to provide a world-class building to house the V&A Museum of Design Dundee.
“Our city’s profile is already benefiting from our association with the V&A and the breathtaking vision of Kengo Kuma’s building. Dundee’s ongoing and ambitious regeneration will reach another level when the construction phase begins.”
Douglas Keillor, Regional Director for BAM Construction in Scotland, said, “This is a project of international significance and it is great to hear that we have been recommended as the preferred main contractor.
“We keenly await the decision of Dundee City Council on Monday.”
The original design to launch half the building across the water has been modified to save costs founding it mainly on the banks of the river Tay with a small element jutting out across the water.
The building envelope will be built as a reinforced concrete structural wall clad in reconstituted precast units externally and timber cladding internally, creating an organic, layered and textured form.