Manchin and Raines said they've seen bids come in all over the board this year, and they're "cautiously optimistic" the figures will be favorable.
The project has been placed on agenda for the authority's Monday meeting in case the bids come in over budget.
The building was designed with the curriculum in mind, he said.
"I have balked at scaling this project back," Manchin said. "If we're going to build a unique building, we need to build it."
Wilson said about 30 contractors were at the project's pre-bid meeting. A contract could be awarded to one company or to multiple subcontractors. Wilson and Raines said bids from "multiple prime contractors" typically come in lower than others.
Manchin said authority officials will meet with county officials Friday to review the bids and perhaps even decide which should be accepted.
Once a bid is selected, contractors need to wait until the building pad is ready, Wilson said.
Adam Krason, an architect with a firm assisting with the project, said the pad is still on schedule to be completed in November. Wilson said the site preparation work then could move to other parts of the site while construction starts on the building.
Construction should take 12 to 14 months, Raines said.
That timeline could change significantly if bidding doesn't go as planned.
"We're praying that the numbers come in good tomorrow," Manchin said Wednesday.
Bids are due by 2 p.m. today at the Kanawha County School's Crede Operations Facility.