The phenomenon of multi-stable synchronization manifold (SM) in generalized synchronization (GS) refers to the coexistence of multiple chaotic response attractors, which all synchronize with the same driving dynamics in the sense of GS. In this paper, the multivalue characteristic of SM is investigated in the general sense on the basis of establishing the functional relationship between the driving and response systems. The stability of SM is studied and the conditions ensuring the existence of the multi-stable SM are deduced. A Genesio-Rssler coupled system and a coupled Duffing system with quadric and cubic nonlinear terms are analyzed as examples and the results show that there exist only one stable SM in the former, while the bifurcation evolves from bistable SM to single-stable SM with the increase of coupling strength in the latter.