Network Rail is expected to officially confirm the deal in two weeks after the contract award standstill period has passed for appeals from rival bidders Bam Nuttall, Carillion and Morgan Sindall.
Work on the project, which is understood to be worth up to £70m, and can now get underway early next year.
The huge railway flyover project forms part of the Reading station redevelopment and will allow mainline passenger services to travel over the top of freight trains up to a mile long coming up from the south.