Skanska had considered entering the race for the arts centre, but Bovis Lend Lease said it had not as it was "the wrong kind of contract".
Construction managers Schal and a Citex/Mace joint venture were previously shortlisted for the job. They stood down in April when the Welsh assembly decided to switch to a lump sum procurement method.
The assembly then set a fixed price of £92m and warned contractors that the entire project would be scrapped if this was exceeded. Of this sum construction costs are capped at £60m.
Last year, contractor Amec's final offer to design and build the centre for £86m plus £10m contingencies was rejected because it exceeded the budget, which was then £75m.
However, a source close to the project said the same problem should not occur again, as every aspect of the scheme and its resulting price had been scrutinised.
He added: "We realise this is a very important and high-profile project, and certainly more difficult than the average office building. But there is nothing pioneering about it.
"There will be a certain amount of strife ahead and it will be understandable for them [the preferred contractor] to revisit the cost control and value engineering decisions. But it's unnecessary as it's been done twice before."
A source at one of the shortlisted contractors admitted firms had been put off the scheme following contractor Laing's problems with the Welsh Rugby Union's Millennium Stadium, also in Cardiff.
The source said: "We certainly don't want to catch a cold on this. But we are reasonably confident we have robust measures in place to ensure this doesn't happen."
The 32,000 m2 project includes an opera house, a recording studio and a theatre.
Once a preferred bidder is selected, the contractor will work through the designs with architect Percy Thomas Partnership and structural engineer Arup. Citex is the quantity surveyor.
Work is due to start in January next year.