The magnetostriction coefficient and its stress dependence are evaluated in Co- and Fe-Ni- based glass-coated microwires with different character of hysteresis loops and positive and low negative magnetostriction respectively. We found that the stress dependence of the magnetostriciton coefficient, λs, is linear for both families of studied microwires, however a decrease with applied stress is observed for Co-rich microwires with negative magnetostriction coefficient. In contrast, a linear increase of the magnetostriction coefficient with applied stress is observed for Fe-Ni- rich microwires with positive magnetostriction coefficient. Observed results are discussed in terms of influence of internal stresses and short range ordering induced by annealing.