In this paper, new multiplicative and additive two-level methods are being proposed, for second kind variational inequalities and quasi-variational inequalities. Section 1 has a preliminary character. Section 2 is devoted to the introduction of a general framework in a reflexive Banach space. It essentially consists of two hypotheses – one for the multiplicative algorithm and the other for the additive algorithm – related to the decomposition of the elements in the convex set. In Section 3, the subspace correction algorithms are defined for second kind variational inequalities and a globally convergence result concerning these is formulated. Further, in Section 4, the subspace correction algorithms for the quasi-variational inequalities are introduced and their convergence properties are being studied, together with a convergence rate estimation. Finally, Section 5 is devoted to the two-level methods; precisely, it is shown that the assumptions introduced in the preceding sections hold for two-obstacle convex sets. Further aspects occasioned by these developments are also being considered.