Under the agreement, White joined Laing last month without having to serve out his notice, but in return has made himself available to help on a number of McAlpine projects.
City sources described the arrangement as "highly unusual" but both companies played this down, saying the deal made sense.
White said the amount of work he was doing for McAlpine was minimal and "for just a few projects". McAlpine is not paying White, who had been head of its capital projects division.
Whitehead said: "Andy has been involved in many things for us for a long time. If I need to talk to Andy then I will. Ray and I have agreed that."
O'Rourke said White was working full time at Laing. "He's working 65 hours, probably 70, for us so what he does outside that is his business."
n Laing Group, which sold Laing Construction to O'Rourke last year, has hinted in a statement at its annual meeting on Wednesday that it may sell its housing business, Laing Homes.