The authority wants to build three 'one-stop shop' leisure centres.
'The way we deliver services is going to change. We are trying to get people on board via public meetings and roadshows,' said Greenwich press officer Michael Diop.
The £20m Eltham Centre will be the first to be built, replacing Eltham Pools with a modern building boasting three swimming pools, a library and an education centre.
Similar centres will be built in Woolwich.