The magnetoimpedance effect has been measured in as-received and current annealed Fe71Cr7Si9B13 amorphous wires under torsional stress varied up to 1.2 rad/cm. It was found that the torsional stress dependence of impedance of as-received amorphous wire has a non-monotonous shape with first an increase of the impedance and then a decrease. On the other hand, the impedance of current annealed wire directly starts to decrease with increasing torsional stress. The current annealing of wire results in a large variation in the magnitude of torsional stress impedance effect, (DeltaZ/Z)xi(%), up to 150%. Also, a simple mathematical model based on the magnetic moment rotation to explain torsional stress versus impedance data was developed and the results were partly discussed on the basis of this model.