Glass-coated amorphous microwires of two different compositions, Co₆₈Mn₇Si₁₀B₁₅ and Fe₇₀B₁₅Si₁₀C₅, with square hysteresis loops have been prepared and studied. The effect of applied tensile stress on their magnetic properties is reported for both kinds of microwires with radii of the metallic nucleus in the range from 1.9-7.5 μm and thicknesses of the coating of 3.8-10.8 μm. The switching field of the studied microwires depends on the sample's geometry (the radius of the metallic nucleus and the thickness of the glass coating), exhibiting a decrease with decreasing thickness of the coating owing to the induced internal stresses. For Fe-rich compositions an increase of the switching field with the applied tensile stress in the range of 0-750 MPa was observed, such an increase being more significant for a thin coating. The classical square root dependence of the switching field on the applied stress is found for Fe-rich compositions for the range of thicker metallic nuclei. Co-rich microwires exhibit quite a different stress dependence of the switching field, exhibiting first roughly stress-independent behaviour and then an increase with the stress. Removal of the glass coating results in a drastic change of the switching fields for both compositions, indicating the existence of strong internal frozen-in stresses. The observed experimental dependences were explained in terms of a dependence of the nucleation field both on external and on internal stresses and by taking into account the dependence of the magnetostriction constant on the strength of the total stresses.