We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of Rn (n 3, p[2,n)) periodically perforated by balls of radius O(epsilon) where >1 and epsilon is the size of the period. The perforations are distributed along a (n-1)-dimensional manifold , 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 epsilon-, R and epsilon is a small parameter that we shall make to go to zero. We analyze different relations between the parameters p, and , and obtain homogenized problems which are completely new in the literature even for the case p=2.