In this paper we propose a local projection stabilized Lagrange multiplier method in order to approximate the two-dimensional linear elastostatics unilateral contact problem with Tresca friction in the framework of the eXtended Finite Element Method X-FEM. This last method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. The advantage of the used stabilization technique is to affect only the equation on multipliers and thus to be equation independent. We study the existence, uniqueness and a priori error estimate of three hybrid discrete formulations.