Willmott Dixon now work with the council on the final stages of design before plans are submitted for planning permission in the coming weeks.
Bransbury Park Leisure Centre will include a general practice medical facility as well as a sports centre with swimming pool.
Project cost was put at £16m at the start of 2022.
The aspiration is to get planning cleared by January and start site preparation in February, enabling main construction to begin next August 2024 and complete in January 2026.
Willmott Dixon southern regional managing director Richard Poulter said: “We have a long track-record of delivering property that brings together a number of uses and services under one roof…. Our focus will now be on developing Bransbury Park Leisure Centre’s plans further so this exciting project can obtain planning permission later this year.”
Willmott Dixon was selected via the Southern Construction Framework.