Purpose. To solve the problem of the allocation of contact vertical normal tensions along the sole of a rigid round stamp, located in an elastic isotropic half-space at a certain depth h  0. To compare the obtained solution with the well-known classical result for h = 0, to check the obtained results for adequacy. Methodology. Based on the analysis of the decision on the stress-strain state of the base, inside which there is a vertical arbitrary load distributed over the area of the circle, the necessary formulas are obtained to solve the problem. An algorithm for constructing an approximate solution has been developed, the essence of which is to use a combination of the boundary element method and the iteration process by S.N.Klepikov. For a number of depths, approximate solutions of the considered problem are obtained. Findings. The proposed algorithm for the approximate solution of the problem of indenting a round rigid stamp into the upper boundary of an elastic isotropic half-space has good agreement with the exact solution and can be used to solve contact problems. The outlines of the contact stress diagrams depend on the depth at which they are determined – the greater the depth, the flatter the outlines of the diagrams are, while starting from a certain depth, the diagrams of contact stresses practically coincide. The greater the depth is at which the stamp is located, the more force must be applied to obtain equal displacements of the stamp. Originality. The obtained research results significantly expand the possibilities of solving various problems of soil mechanics and foundation engineering, make it possible to obtain absolutely new results. In particular, a clear dependence of the contact stresses along the sole of a rigid round stamp on the depth at which it is located was identified. In addition, the presented data allow us to designate an absolutely new direction in the calculation of the foundations of ground anchors, namely, the calculation of their deformations. Practical value. For engineering practice, it is important that the greater the value of Poisson’s ratio of the base is, the greater the contact stresses are, other things being equal.