The computational study of discontinuous problems becomes increasingly important due to industrial needs, such as the application of natural or artificial composites and the understanding of damage and fracture processes of materials. The discontinuity-enriched finite element method (DE-FEM) and interface-enriched generalized finite element method (IGFEM) are promising computational approaches developed in recent years for solving problems containing discontinuities, which are capable of decoupling the mesh morphology from the geometry of discontinuities. Nonetheless, these formulations have only been developed for linear or quadratic polynomial interpolants. We introduce high-order discontinuity formulations aiming at improving the accuracy of the approximation by fixing the mesh size while increasing the polynomial degree of the interpolant, namely the p-version of DE-FEM/IGFEM (p-DE-FEM/p-IGFEM). In the proposed methods, hierarchical shape functions of the p-version of FEM (p-FEM) are employed, and high-order enrichment functions constructed with different polynomial bases are applied in elements intersected by discontinuities. Through convergence tests, we show that p-DE-FEM solves one-dimensional elasticity problems with exponential rates of convergence, and p-IGFEM solves two-dimensional static heat conduction problems with the same exponential rates of convergence as those of p-FEM with meshes that align to discontinuities. When the mesh is refined and the polynomial degree is fixed, p-IGFEM obtains optimal rates of O(p), with p denoting the polynomial degree used. We also show that high-order enrichment functions constructed with orthogonal polynomial bases lead to better-conditioned stiffness matrices.