They are Pools on the Park, Briar House, East Sheen Library, Orleans House Gallery, and Teddington Swimming Pool.
The Council is also committing £1.94 million in match funding to help deliver these ambitious plans.
The funding is coming from Salix Finance on behalf of the Department for Energy Security and Net Zero.
The improvements will include low-carbon heating systems and upgraded insulation.
It’s expected the move will cut carbon emissions by an estimated 951 tonnes of CO2 every year.