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We studied the magnetostriction coefficients, <i>λ_s</i>, Curie temperature, <i>T_c</i>, and their dependence on annealing conditions in Fe₄₇Ni₂₇Si₁₁B₁₃C₂ and Co₆₇Fe_3.9Ni₁.₅B₁₁.₅Si_14.5Mo_1.6 amorphous glass-coated microwires with rather different character of hysteresis loops. A positive <i>λ_s</i> ≈ 20 × 10⁻⁶ is observed in as-prepared Fe₄₇Ni₂₇Si₁₁B₁₃C_2, while low and negative <i>λ_s</i> ≈ −0.3 × 10⁻⁶ is obtained for Co₆₇Fe_3.9Ni₁.₅B₁₁.₅Si_14.5Mo_1.6 microwire. Annealing affects the magnetostriction coefficients and Curie temperatures, <i>T_c</i>, of both Fe₄₇Ni₂₇Si₁₁B₁₃C₂ and Co₆₇Fe_3.9Ni₁.₅B₁₁.₅Si_14.5Mo_1.6 glass-coated microwires in a similar way. Observed dependencies of hysteresis loops, <i>λ_s</i> and <i>T_c</i> on annealing conditions are discussed in terms of superposition of internal stresses relaxation and structural relaxation of studied microwires. We observed linear <i>λ_s</i> dependence on applied stress, σ, in both studied microwires. A decrease in the magnetostriction coefficient upon applied stress is observed for Co-rich microwires with low and negative magnetostriction coefficient. On the contrary, for Fe-Ni-rich microwires with a positive magnetostriction coefficient, an increase in the magnetostriction coefficient with applied stress is observed. The observed results are discussed considering the internal stresses relaxation and short range atomic rearrangements induced by annealing on hysteresis loops, magnetostriction coefficients and Curie temperatures of studied microwires.