The new centre will be located on Bakers Lane car park site and will provide a six-lane swimming pool, fitness suite, gym, studios, sports hall and two squash courts.
Plans for the new Epping Leisure Centre were first approved by Epping Forest District Council in October 2021. Since then the project estimate has risen by £8m to £33m.
The South London-based builder is also preferred bidder to develop a £13m leisure centre at Whitchurch in Shropshire.
The firm is preparing to demolish the existing Whitchurch centre ahead of securing planning for a six-lane 25-metre pool, 41-station gym, two fitness studios and a 20-cover cafe.