The overall budget for the national arts centre is £92m, and the ceiling on construction costs is £60m.
The centre expects to sign off the contract with McAlpine by the end of the year.
A project spokesperson said: "Now we have to go through everything with a fine-tooth comb to get the final fixed price."
The choice of McAlpine is the latest development in the troubled history of the centre.
The original plans for the project were scrapped last year, after Amec and the centre could not agree on a price. In April, construction managers Schal and Citex/Mace were stood down after they refused to give guarantees about the price.