Wakefield Council first revealed proposals for the new facility in Pontefract Park in 2016, when it announced cost-cutting plans to shut the now-closed Castleford and Knottingley swimming pools.
At the time, it said the complex would cost Â£14.8m.
But next week, members of the council’s cabinet will instead be asked to approve a budget of up to Â£20.8m for the scheme.
The authority said Â£2.2m would be “external” funding, but this has yet to be finalised.
Coun Jacquie Speight, cabinet member for culture, leisure and sport, said: “This is an exciting project and we’ll be discussing how the development plans have progressed since the feasibility study.
“Our aim is to give people in the Five Towns area the opportunity to enjoy first class leisure facilities in a new and modern building.”
The proposed centre would include a 10-lane swimming pool, gym and exercise studios, a clip and climb activity and cafe.
It is hoped the centre will open in summer 2020.