The firm is said to be in detailed talks with the engineering giant to deliver a factory building large pressure vessels for the next generation of nuclear power stations at the Advanced Manufacturing Park.
The deal will be a coup for Bam, which until three months ago had not worked for Rolls-Royce.
Bam Construction won its first contract with Rolls Royce to build the new blade casting factory in Rothertham earlier this summer.
A source told the Enquirer: “There’s been no confirmation but it looks like Bam could take the Project PoWer factory job as well.”
The latest factory is slightly larger in size and estimated to cost around £30m to build.
Construction is scheduled to begin this autumn and it is anticipated that the first of the pressure vessels would be produced in 2015.
Rotherham Council last week granted approval for the 226,000 sq ft factory building, which will rise to 33m at its highest point to accommodate the 23m high pressure vessels.