McAlpine led Team Stadium which also included Populous as the Architect and Buro Happold as the designer of the civils, structural and building services work on the 2012 stadium.
Populous is already on board as architect for the shell-and-core conversion contract.
Bidders who have expressed an interest in becoming main contractor are believed to include McAlpine, Balfour Beatty, Buckingham, ISG and Shepherd.
One source close to the bidding told the Enquirer: “Obviously McAlpine are in a strong position because they built the original stadium.
“But the Olympic authorities run very fair procurement races so everyone else going for this will give it a real go and it won’t be a walkover for McAlpine.”
A winner for the race to convert the 2012 centre piece into a 54,000-seater stadium for West Ham is due to be chosen by the end of this year.