Rectangular hysteresis loops have been observed in Co₆₈Mn₇Si₁₀B₁₅ amorphous glass-coated microwires of different geometry with ratio r of metallic nucleus radius to total radius of the microwires ranging between 0.62 and 0.17. Switching field H* decreases from 200 to 40 A/m when r increases from 0.17 to 0.62. Dependence of H* on external tensile stresses, σ_a, with different r was studied in as-prepared state and after glass-coating removal by chemical etching in HF. H* increases with σ for each sample after certain roughly stress-independent region. Chemical etching generally results in a decrease of H*. Dependence of magnetostriction constant on the strength of the total external and internal stresses was taken into account to explain the experimental results.