The contractor was told on Friday it had won the 12,000 seat arena project, subject to the usual 10-day standstill period, according to Enquirer sources.
It marks a major coup for Bouygues in the south west, after beating rival bids from BAM Construction, Buckingham Group, Laing O’Rourke and Sir Robert McAlpine.
International arena design team Populous together with Bath-based architects Feilden Clegg Bradley and engineering firm Buro Happold have developed plans for the innovative and flexible arena.
It will be financed by the council borrow £53m against its City Deal and raising the remaining £38m from the arena operator and car parking income.
According to city council documents the Arena is now due to open in early 2018.