Lovász and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct vertices in $ T $ is $ \sum_{t\in T}λ(t, T-t) $ where $λ$ is the local edge-connectivity function. The optimality of a system of edge-disjoint $ T $-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs.