The Midlands based contractor is understood to have edged out rivals Bowmer & Kirkland and Vinci Construction in the chase, a source told the Enquirer.
The job is one of the biggest industrial jobs to be awarded this year.
Preparatory works are already underway at the i54 business park with Bam Nuttall carrying out a major earthworks package.
During the next two months Bam will cut and fill up to 1m cu m of soil ahead of building work starting.
When finished the new 800,000 sq ft factory will provide 300,000 low emission, four-cylinder petrol and diesel engines each year, supporting 3,000 supply chain jobs.
Meanwhile, Jaguar Land Rover yesterday unveiled plans to invest £200m expanding its plant at Castle Bromwich.
The factory will be extended by 50% to build the new Jaguar F-Type sports car, the successor to the historic E-type.