Within the general approach to the modified Kirsch problem incorporating surface elasticity and residual surface stress (surface tension) by the original Gurtin–Murdoch model, the proper boundary conditions at an arbitrary cylindrical surface are derived in terms of complex variables for the plane strain and plane stress. In the case of the plane stress, the properties are allowed for not only of the cylindrical surface but the faces of a plate with the nanosized thickness as well. The solution of both 2-D problems on the infinite plane with the circular hole subjected to a uniform far-field load is obtained in a closed form. It is shown that under plane strain conditions, the general formulas for the stress tensor components are reduced to the modified Kirsch solution at the nanoscale in the case of the uniaxial loading. At the same time, the uniaxial remote loading alone is impossible in the case of the plane stress because of the presence of the axisymmetric surface tension at the faces of the plate. Analytical solution shows that the elastic field of the plate depends on the plate thickness. Numerical examples are given in the paper to illustrate quantitatively the effect of the plate thickness on the stress field at the cylindrical surface and the role of surface tension in the corresponding plane strain problem.