The firm plans to double revenue in the region to £600 million by 2009. It carries out work in Egypt, Abu Dhabi and Dubai, plus neighbouring Oman.
Carillion group finance director Richard Adam said it has also been asked by Qatar’s state government to look into carrying out work. He added: “Their infrastructure spend is going to be huge.”
Mr Greenwood is the only member of the Alfred McAlpine board to stay on at the group.
Former chief executive Ian Grice has left with a £5.2 million pay-off while project services director Matt Swan and Craig McGilvray, who headed its business services arm, have also gone.
Mr Adam confirmed it will not be using the Alfred McAlpine name in the future, although environmental consultant Enviros, bought by McAlpine last year, will keep its name.
He said the firm expects to make synergy cost savings of £30 million a year from 2009.
Carillion saw pre-tax profits last year jump 39 per cent to £94.4 million and revenue rise 13 per cent to £3.9 billion.