The development of carbon-based reverse osmosis membranes for water desalination is hindered by challenges in achieving a high pore density and controlling the pore size. We use molecular dynamics simulations to demonstrate that graphene foam membranes with a high pore density provide the possibility to tune the pore size by applying mechanical strain. As the pore size is found to be effectively reduced by a structural transformation under strain, graphene foam membranes are able to combine perfect salt rejection with unprecedented water permeability.