The Olympic Delivery Authority (ODA) has issued an OJEU contract notice for constructing the east London venue.
It is also inviting bids for several unconfirmed projects, including a temporary water-polo venue, landscaping and a permanent plaza.
Hadid's scheme has been radically overhauled since first being unveiled. It now has a roof just over a third of the size of the original proposals - the new structure will cover 14,000m2 compared to the 35,000m2 envisaged in the competition-winning design. (Zaha reveals radical changes to Olympic Aquatic Centre).
Another loss is the unusual wave-like dip in the roof form, which was a key characteristic of the original design.
Launching the construction competition, ODA boss David Higgins said: 'This is an opportunity for a major construction company to build a world-class London 2012 venue fit for the best athletes in the world and a lasting legacy.'
He added: 'The Aquatics Centre will form a spectacular gateway to the Olympic Park from Stratford. After the Games it will become a new landmark for London, a new home for UK swimming and a new swimming and fitness facility for local people.'
Work is already under way clearing and excavating the site. Construction is scheduled to begin in 2008 in time for training and test events in 2011.
Culture Secretary Tessa Jowell has announced that the budget for the 2012 Olympic Games has risen to £9.3 billion. Construction costs are now estimated at £5.3 billion - up from an initial estimate of £2.4 billion.