We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of R n (n ≥ 3, p ∈ [2, n)) periodically perforated by balls of radius O(ε α) where α > 1 and ε is the size of the period. The perforations are distributed along a (n − 1)dimensional manifold γ, and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter ε −κ , κ ∈ R and ε is a small parameter that we shall make to go to zero. We analyze different relations between the parameters p, n, ε, α and κ, and obtain homogenized problems which are completely new in the literature even for the case p = 2.