Plans for the Fleming Park leisure centre in Eastleigh include a larger swimming pool with eight lanes and a larger learner pool with a moveable floor.
The sports complex will also contain a 15-court hall, a large gym and four outdoor tennis courts.
A contractor is due to be selected in August with a target completion date of June 2017.
Firms have until the 19 June to complete prequals, which are available from the council website.